class Dokchitser(SageObject): r""" Dokchitser's `L`-functions Calculator Create a Dokchitser `L`-series with Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec) where - ``conductor`` - integer, the conductor - ``gammaV`` - list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples). - ``weight`` - positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for `H^1` of curves/`\QQ` - ``eps`` - complex number; sign in functional equation - ``poles`` - (default: []) list of points where `L^*(s)` has (simple) poles; only poles with `Re(s)>weight/2` should be included - ``residues`` - vector of residues of `L^*(s)` in those poles or set residues='automatic' (default value) - ``prec`` - integer (default: 53) number of *bits* of precision RIEMANN ZETA FUNCTION: We compute with the Riemann Zeta function. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L Dokchitser L-series of conductor 1 and weight 1 sage: L(1) Traceback (most recent call last): ... ArithmeticError sage: L(2) 1.64493406684823 sage: L(2, 1.1) 1.64493406684823 sage: L.derivative(2) -0.937548254315844 sage: h = RR('0.0000000000001') sage: (zeta(2+h) - zeta(2.))/h -0.937028232783632 sage: L.taylor_series(2, k=5) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5) RANK 1 ELLIPTIC CURVE: We compute with the `L`-series of a rank `1` curve. :: sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser(); L Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: L(1) 0.000000000000000 sage: L.derivative(1) 0.305999773834052 sage: L.derivative(1,2) 0.373095594536324 sage: L.num_coeffs() 48 sage: L.taylor_series(1,4) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4) sage: L.check_functional_equation() 6.11218974700000e-18 # 32-bit 6.04442711160669e-18 # 64-bit RANK 2 ELLIPTIC CURVE: We compute the leading coefficient and Taylor expansion of the `L`-series of a rank `2` curve. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs () 156 sage: L.derivative(1,E.rank()) 1.51863300057685 sage: L.taylor_series(1,4) -1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit -2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit RAMANUJAN DELTA L-FUNCTION: The coefficients are given by Ramanujan's tau function:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) We redefine the default bound on the coefficients: Deligne's estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster). :: sage: L.num_coeffs() 12 sage: L.set_coeff_growth('2*n^(11/2)') sage: L.num_coeffs() 11 Now we're ready to evaluate, etc. :: sage: L(1) 0.0374412812685155 sage: L(1, 1.1) 0.0374412812685155 sage: L.taylor_series(1,3) 0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3) """ def __init__(self, conductor, gammaV, weight, eps, \ poles=[], residues='automatic', prec=53, init=None): """ Initialization of Dokchitser calculator EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """ self.conductor = conductor self.gammaV = gammaV self.weight = weight self.eps = eps self.poles = poles self.residues = residues self.prec = prec self.__CC = ComplexField(self.prec) self.__RR = self.__CC._real_field() if not init is None: self.init_coeffs(init) self.__init = init else: self.__init = False def __reduce__(self): D = copy.copy(self.__dict__) if '_Dokchitser__gp' in D: del D['_Dokchitser__gp'] return reduce_load_dokchitser, (D, ) def _repr_(self): z = "Dokchitser L-series of conductor %s and weight %s"%( self.conductor, self.weight) return z def __del__(self): self.gp().quit() def gp(self): """ Return the gp interpreter that is used to implement this Dokchitser L-function. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L(2) 0.546048036215014 sage: L.gp() PARI/GP interpreter """ try: return self.__gp except AttributeError: logfile = None # For debugging import os from sage.env import DOT_SAGE logfile = os.path.join(DOT_SAGE, 'dokchitser.log') g = sage.interfaces.gp.Gp(script_subdirectory='dokchitser', logfile=logfile) g.read('computel.gp') self.__gp = g self._gp_eval('default(realprecision, %s)'%(self.prec//3 + 2)) self._gp_eval('conductor = %s'%self.conductor) self._gp_eval('gammaV = %s'%self.gammaV) self._gp_eval('weight = %s'%self.weight) self._gp_eval('sgn = %s'%self.eps) self._gp_eval('Lpoles = %s'%self.poles) self._gp_eval('Lresidues = %s'%self.residues) g._dokchitser = True return g def _gp_eval(self, s): try: t = self.gp().eval(s) except (RuntimeError, TypeError): raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.") if '***' in t: raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.") return t def __check_init(self): if not self.__init: raise ValueError("you must call init_coeffs on the L-function first") def num_coeffs(self, T=1): """ Return number of coefficients `a_n` that are needed in order to perform most relevant `L`-function computations to the desired precision. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 26 sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 568 sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """ return Integer(self.gp().eval('cflength(%s)'%T)) def init_coeffs(self, v, cutoff=1, w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40): """ Set the coefficients `a_n` of the `L`-series. If `L(s)` is not equal to its dual, pass the coefficients of the dual as the second optional argument. INPUT: - ``v`` - list of complex numbers or string (pari function of k) - ``cutoff`` - real number = 1 (default: 1) - ``w`` - list of complex numbers or string (pari function of k) - ``pari_precode`` - some code to execute in pari before calling initLdata - ``max_imaginary_part`` - (default: 0): redefine if you want to compute L(s) for s having large imaginary part, - ``max_asymp_coeffs`` - (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t). EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) Evaluate the resulting L-function at a point, and compare with the answer that one gets "by definition" (of L-function attached to a modular form):: sage: L(14) 0.998583063162746 sage: a = delta_qexp(1000) sage: sum(a[n]/float(n)^14 for n in range(1,1000)) 0.9985830631627459 Illustrate that one can give a list of complex numbers for v (see :trac:`10937`):: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:]) sage: L2(14) 0.998583063162746 TESTS: Verify that setting the `w` parameter does not raise an error (see :trac:`10937`). Note that the meaning of `w` does not seem to be documented anywhere in Dokchitser's package yet, so there is no claim that the example below is meaningful! :: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000]) """ if isinstance(v, tuple) and w is None: v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v self.__init = (v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs) gp = self.gp() if pari_precode != '': self._gp_eval(pari_precode) RR = self.__CC._real_field() cutoff = RR(cutoff) if isinstance(v, str): if w is None: self._gp_eval('initLdata("%s", %s)'%(v, cutoff)) return self._gp_eval('initLdata("%s",%s,"%s")'%(v,cutoff,w)) return if not isinstance(v, (list, tuple)): raise TypeError("v (=%s) must be a list, tuple, or string"%v) CC = self.__CC v = ','.join([CC(a)._pari_init_() for a in v]) self._gp_eval('Avec = [%s]'%v) if w is None: self._gp_eval('initLdata("Avec[k]", %s)'%cutoff) return w = ','.join([CC(a)._pari_init_() for a in w]) self._gp_eval('Bvec = [%s]'%w) self._gp_eval('initLdata("Avec[k]",%s,"Bvec[k]")'%cutoff) def __to_CC(self, s): s = s.replace('.E','.0E').replace(' ','') return self.__CC(sage_eval(s, locals={'I':self.__CC.gen(0)})) def _clear_value_cache(self): del self.__values def __call__(self, s, c=None): r""" INPUT: - ``s`` - complex number .. note:: Evaluation of the function takes a long time, so each evaluation is cached. Call ``self._clear_value_cache()`` to clear the evaluation cache. EXAMPLES:: sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser(100) sage: L(1) 0.00000000000000000000000000000 sage: L(1+I) -1.3085436607849493358323930438 + 0.81298000036784359634835412129*I """ self.__check_init() s = self.__CC(s) try: return self.__values[s] except AttributeError: self.__values = {} except KeyError: pass z = self.gp().eval('L(%s)'%s) if 'pole' in z: print(z) raise ArithmeticError elif '***' in z: print(z) raise RuntimeError elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits','decimal digits') verbose(msg, level=-1) ans = self.__to_CC(z[i+1:]) self.__values[s] = ans return ans ans = self.__to_CC(z) self.__values[s] = ans return ans def derivative(self, s, k=1): r""" Return the `k`-th derivative of the `L`-series at `s`. .. warning:: If `k` is greater than the order of vanishing of `L` at `s` you may get nonsense. EXAMPLES:: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.derivative(1,E.rank()) 1.51863300057685 """ self.__check_init() s = self.__CC(s) k = Integer(k) z = self.gp().eval('L(%s,,%s)'%(s,k)) if 'pole' in z: raise ArithmeticError(z) elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits','decimal digits') verbose(msg, level=-1) return self.__CC(z[i:]) return self.__CC(z) def taylor_series(self, a=0, k=6, var='z'): r""" Return the first `k` terms of the Taylor series expansion of the `L`-series about `a`. This is returned as a series in ``var``, where you should view ``var`` as equal to `s-a`. Thus this function returns the formal power series whose coefficients are `L^{(n)}(a)/n!`. INPUT: - ``a`` - complex number (default: 0); point about which to expand - ``k`` - integer (default: 6), series is `O(``var``^k)` - ``var`` - string (default: 'z'), variable of power series EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.taylor_series(2, 3) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3) sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser() sage: L.taylor_series(1) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6) We compute a Taylor series where each coefficient is to high precision. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(200) sage: L.taylor_series(1,3) -9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) """ self.__check_init() a = self.__CC(a) k = Integer(k) try: z = self.gp()('Vec(Lseries(%s,,%s))'%(a,k-1)) except TypeError as msg: raise RuntimeError("%s\nUnable to compute Taylor expansion (try lowering the number of terms)"%msg) r = repr(z) if 'pole' in r: raise ArithmeticError(r) elif 'Warning' in r: i = r.rfind('\n') msg = r[:i].replace('digits','decimal digits') verbose(msg, level=-1) v = list(z) K = self.__CC v = [K(repr(x)) for x in v] R = self.__CC[[var]] return R(v,len(v)) def check_functional_equation(self, T=1.2): r""" Verifies how well numerically the functional equation is satisfied, and also determines the residues if ``self.poles != []`` and residues='automatic'. More specifically: for `T>1` (default 1.2), ``self.check_functional_equation(T)`` should ideally return 0 (to the current precision). - if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong), - if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata("a(k)",1.3), checkfeq(1.3) - T=1 always (!) returns 0, so T has to be away from 1 - default value `T=1.2` seems to give a reasonable balance - if you don't have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata("a(k)",1), you need slightly less coefficients. EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -1.35525271600000e-20 # 32-bit -2.71050543121376e-20 # 64-bit If we choose the sign in functional equation for the `\zeta` function incorrectly, the functional equation doesn't check out. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -9.73967861488124 """ self.__check_init() z = self.gp().eval('checkfeq(%s)'%T).replace(' ','') return self.__CC(z) def set_coeff_growth(self, coefgrow): r""" You might have to redefine the coefficient growth function if the `a_n` of the `L`-series are not given by the following PARI function:: coefgrow(n) = if(length(Lpoles), 1.5*n^(vecmax(real(Lpoles))-1), sqrt(4*n)^(weight-1)); INPUT: - ``coefgrow`` - string that evaluates to a PARI function of n that defines a coefgrow function. EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) sage: L.set_coeff_growth('2*n^(11/2)') sage: L(1) 0.0374412812685155 """ if not isinstance(coefgrow, str): raise TypeError("coefgrow must be a string") g = self.gp() g.eval('coefgrow(n) = %s'%(coefgrow.replace('\n',' ')))
class Dokchitser(SageObject): r""" Dokchitser's `L`-functions Calculator Create a Dokchitser `L`-series with Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec) where - ``conductor`` - integer, the conductor - ``gammaV`` - list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples). - ``weight`` - positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for `H^1` of curves/`\QQ` - ``eps`` - complex number; sign in functional equation - ``poles`` - (default: []) list of points where `L^*(s)` has (simple) poles; only poles with `Re(s)>weight/2` should be included - ``residues`` - vector of residues of `L^*(s)` in those poles or set residues='automatic' (default value) - ``prec`` - integer (default: 53) number of *bits* of precision RIEMANN ZETA FUNCTION: We compute with the Riemann Zeta function. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L Dokchitser L-series of conductor 1 and weight 1 sage: L(1) Traceback (most recent call last): ... ArithmeticError sage: L(2) 1.64493406684823 sage: L(2, 1.1) 1.64493406684823 sage: L.derivative(2) -0.937548254315844 sage: h = RR('0.0000000000001') sage: (zeta(2+h) - zeta(2.))/h -0.937028232783632 sage: L.taylor_series(2, k=5) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5) RANK 1 ELLIPTIC CURVE: We compute with the `L`-series of a rank `1` curve. :: sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser(); L Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: L(1) 0.000000000000000 sage: L.derivative(1) 0.305999773834052 sage: L.derivative(1,2) 0.373095594536324 sage: L.num_coeffs() 48 sage: L.taylor_series(1,4) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4) sage: L.check_functional_equation() 6.11218974700000e-18 # 32-bit 6.04442711160669e-18 # 64-bit RANK 2 ELLIPTIC CURVE: We compute the leading coefficient and Taylor expansion of the `L`-series of a rank `2` curve. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs () 156 sage: L.derivative(1,E.rank()) 1.51863300057685 sage: L.taylor_series(1,4) -1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit -2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit RAMANUJAN DELTA L-FUNCTION: The coefficients are given by Ramanujan's tau function:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) We redefine the default bound on the coefficients: Deligne's estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster). :: sage: L.num_coeffs() 12 sage: L.set_coeff_growth('2*n^(11/2)') sage: L.num_coeffs() 11 Now we're ready to evaluate, etc. :: sage: L(1) 0.0374412812685155 sage: L(1, 1.1) 0.0374412812685155 sage: L.taylor_series(1,3) 0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3) """ def __init__(self, conductor, gammaV, weight, eps, \ poles=[], residues='automatic', prec=53, init=None): """ Initialization of Dokchitser calculator EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """ self.conductor = conductor self.gammaV = gammaV self.weight = weight self.eps = eps self.poles = poles self.residues = residues self.prec = prec self.__CC = ComplexField(self.prec) self.__RR = self.__CC._real_field() if not init is None: self.init_coeffs(init) self.__init = init else: self.__init = False def __reduce__(self): D = copy.copy(self.__dict__) if '_Dokchitser__gp' in D: del D['_Dokchitser__gp'] return reduce_load_dokchitser, (D, ) def _repr_(self): z = "Dokchitser L-series of conductor %s and weight %s"%( self.conductor, self.weight) return z def __del__(self): self.gp().quit() def gp(self): """ Return the gp interpreter that is used to implement this Dokchitser L-function. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L(2) 0.546048036215014 sage: L.gp() PARI/GP interpreter """ try: return self.__gp except AttributeError: logfile = None # For debugging import os from sage.env import DOT_SAGE logfile = os.path.join(DOT_SAGE, 'dokchitser.log') g = sage.interfaces.gp.Gp(script_subdirectory='dokchitser', logfile=logfile) g.read('computel.gp') self.__gp = g self._gp_eval('default(realprecision, %s)'%(self.prec//3 + 2)) self._gp_eval('conductor = %s'%self.conductor) self._gp_eval('gammaV = %s'%self.gammaV) self._gp_eval('weight = %s'%self.weight) self._gp_eval('sgn = %s'%self.eps) self._gp_eval('Lpoles = %s'%self.poles) self._gp_eval('Lresidues = %s'%self.residues) g._dokchitser = True return g def _gp_eval(self, s): try: t = self.gp().eval(s) except (RuntimeError, TypeError): raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.") if '***' in t: raise RuntimeError("Unable to create L-series, due to precision or other limits in PARI.") return t def __check_init(self): if not self.__init: raise ValueError("you must call init_coeffs on the L-function first") def num_coeffs(self, T=1): """ Return number of coefficients `a_n` that are needed in order to perform most relevant `L`-function computations to the desired precision. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 26 sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser() sage: L.num_coeffs() 568 sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """ return Integer(self.gp().eval('cflength(%s)'%T)) def init_coeffs(self, v, cutoff=1, w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40): """ Set the coefficients `a_n` of the `L`-series. If `L(s)` is not equal to its dual, pass the coefficients of the dual as the second optional argument. INPUT: - ``v`` - list of complex numbers or string (pari function of k) - ``cutoff`` - real number = 1 (default: 1) - ``w`` - list of complex numbers or string (pari function of k) - ``pari_precode`` - some code to execute in pari before calling initLdata - ``max_imaginary_part`` - (default: 0): redefine if you want to compute L(s) for s having large imaginary part, - ``max_asymp_coeffs`` - (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t). EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) Evaluate the resulting L-function at a point, and compare with the answer that one gets "by definition" (of L-function attached to a modular form):: sage: L(14) 0.998583063162746 sage: a = delta_qexp(1000) sage: sum(a[n]/float(n)^14 for n in range(1,1000)) 0.9985830631627459 Illustrate that one can give a list of complex numbers for v (see :trac:`10937`):: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:]) sage: L2(14) 0.998583063162746 TESTS: Verify that setting the `w` parameter does not raise an error (see :trac:`10937`). Note that the meaning of `w` does not seem to be documented anywhere in Dokchitser's package yet, so there is no claim that the example below is meaningful! :: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000]) """ if isinstance(v, tuple) and w is None: v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v self.__init = (v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs) gp = self.gp() if pari_precode != '': self._gp_eval(pari_precode) RR = self.__CC._real_field() cutoff = RR(cutoff) if isinstance(v, str): if w is None: self._gp_eval('initLdata("%s", %s)'%(v, cutoff)) return self._gp_eval('initLdata("%s",%s,"%s")'%(v,cutoff,w)) return if not isinstance(v, (list, tuple)): raise TypeError("v (=%s) must be a list, tuple, or string"%v) CC = self.__CC v = ','.join([CC(a)._pari_init_() for a in v]) self._gp_eval('Avec = [%s]'%v) if w is None: self._gp_eval('initLdata("Avec[k]", %s)'%cutoff) return w = ','.join([CC(a)._pari_init_() for a in w]) self._gp_eval('Bvec = [%s]'%w) self._gp_eval('initLdata("Avec[k]",%s,"Bvec[k]")'%cutoff) def __to_CC(self, s): s = s.replace('.E','.0E').replace(' ','') return self.__CC(sage_eval(s, locals={'I':self.__CC.gen(0)})) def _clear_value_cache(self): del self.__values def __call__(self, s, c=None): r""" INPUT: - ``s`` - complex number .. note:: Evaluation of the function takes a long time, so each evaluation is cached. Call ``self._clear_value_cache()`` to clear the evaluation cache. EXAMPLES:: sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser(100) sage: L(1) 0.00000000000000000000000000000 sage: L(1+I) -1.3085436607849493358323930438 + 0.81298000036784359634835412129*I """ self.__check_init() s = self.__CC(s) try: return self.__values[s] except AttributeError: self.__values = {} except KeyError: pass z = self.gp().eval('L(%s)'%s) if 'pole' in z: print(z) raise ArithmeticError elif '***' in z: print(z) raise RuntimeError elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits','decimal digits') verbose(msg, level=-1) ans = self.__to_CC(z[i+1:]) self.__values[s] = ans return ans ans = self.__to_CC(z) self.__values[s] = ans return ans def derivative(self, s, k=1): r""" Return the `k`-th derivative of the `L`-series at `s`. .. warning:: If `k` is greater than the order of vanishing of `L` at `s` you may get nonsense. EXAMPLES:: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser() sage: L.derivative(1,E.rank()) 1.51863300057685 """ self.__check_init() s = self.__CC(s) k = Integer(k) z = self.gp().eval('L(%s,,%s)'%(s,k)) if 'pole' in z: raise ArithmeticError(z) elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits','decimal digits') verbose(msg, level=-1) return self.__CC(z[i:]) return self.__CC(z) def taylor_series(self, a=0, k=6, var='z'): r""" Return the first `k` terms of the Taylor series expansion of the `L`-series about `a`. This is returned as a series in ``var``, where you should view ``var`` as equal to `s-a`. Thus this function returns the formal power series whose coefficients are `L^{(n)}(a)/n!`. INPUT: - ``a`` - complex number (default: 0); point about which to expand - ``k`` - integer (default: 6), series is `O(``var``^k)` - ``var`` - string (default: 'z'), variable of power series EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.taylor_series(2, 3) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3) sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser() sage: L.taylor_series(1) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6) We compute a Taylor series where each coefficient is to high precision. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(200) sage: L.taylor_series(1,3) -9.094...e-82 + (5.1538...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) """ self.__check_init() a = self.__CC(a) k = Integer(k) try: z = self.gp()('Vec(Lseries(%s,,%s))'%(a,k-1)) except TypeError as msg: raise RuntimeError("%s\nUnable to compute Taylor expansion (try lowering the number of terms)"%msg) r = repr(z) if 'pole' in r: raise ArithmeticError(r) elif 'Warning' in r: i = r.rfind('\n') msg = r[:i].replace('digits','decimal digits') verbose(msg, level=-1) v = list(z) K = self.__CC v = [K(repr(x)) for x in v] R = self.__CC[[var]] return R(v,len(v)) def check_functional_equation(self, T=1.2): r""" Verifies how well numerically the functional equation is satisfied, and also determines the residues if ``self.poles != []`` and residues='automatic'. More specifically: for `T>1` (default 1.2), ``self.check_functional_equation(T)`` should ideally return 0 (to the current precision). - if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong), - if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata("a(k)",1.3), checkfeq(1.3) - T=1 always (!) returns 0, so T has to be away from 1 - default value `T=1.2` seems to give a reasonable balance - if you don't have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata("a(k)",1), you need slightly less coefficients. EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -1.35525271600000e-20 # 32-bit -2.71050543121376e-20 # 64-bit If we choose the sign in functional equation for the `\zeta` function incorrectly, the functional equation doesn't check out. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -9.73967861488124 """ self.__check_init() z = self.gp().eval('checkfeq(%s)'%T).replace(' ','') return self.__CC(z) def set_coeff_growth(self, coefgrow): r""" You might have to redefine the coefficient growth function if the `a_n` of the `L`-series are not given by the following PARI function:: coefgrow(n) = if(length(Lpoles), 1.5*n^(vecmax(real(Lpoles))-1), sqrt(4*n)^(weight-1)); INPUT: - ``coefgrow`` - string that evaluates to a PARI function of n that defines a coefgrow function. EXAMPLE:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) sage: L.set_coeff_growth('2*n^(11/2)') sage: L(1) 0.0374412812685155 """ if not isinstance(coefgrow, str): raise TypeError("coefgrow must be a string") g = self.gp() g.eval('coefgrow(n) = %s'%(coefgrow.replace('\n',' ')))
class Dokchitser(SageObject): r""" Dokchitser's `L`-functions Calculator Create a Dokchitser `L`-series with Dokchitser(conductor, gammaV, weight, eps, poles, residues, init, prec) where - ``conductor`` -- integer, the conductor - ``gammaV`` -- list of Gamma-factor parameters, e.g. [0] for Riemann zeta, [0,1] for ell.curves, (see examples). - ``weight`` -- positive real number, usually an integer e.g. 1 for Riemann zeta, 2 for `H^1` of curves/`\QQ` - ``eps`` -- complex number; sign in functional equation - ``poles`` -- (default: []) list of points where `L^*(s)` has (simple) poles; only poles with `Re(s)>weight/2` should be included - ``residues`` -- vector of residues of `L^*(s)` in those poles or set residues='automatic' (default value) - ``prec`` -- integer (default: 53) number of *bits* of precision RIEMANN ZETA FUNCTION: We compute with the Riemann Zeta function. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L Dokchitser L-series of conductor 1 and weight 1 sage: L(1) Traceback (most recent call last): ... ArithmeticError sage: L(2) 1.64493406684823 sage: L(2, 1.1) 1.64493406684823 sage: L.derivative(2) -0.937548254315844 sage: h = RR('0.0000000000001') sage: (zeta(2+h) - zeta(2.))/h -0.937028232783632 sage: L.taylor_series(2, k=5) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 - 1.00002430047384*z^3 + 1.00006193307...*z^4 + O(z^5) RANK 1 ELLIPTIC CURVE: We compute with the `L`-series of a rank `1` curve. :: sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser(algorithm='gp'); L Dokchitser L-function associated to Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: L(1) 0.000000000000000 sage: L.derivative(1) 0.305999773834052 sage: L.derivative(1,2) 0.373095594536324 sage: L.num_coeffs() 48 sage: L.taylor_series(1,4) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + O(z^4) sage: L.check_functional_equation() # abs tol 1e-19 6.04442711160669e-18 RANK 2 ELLIPTIC CURVE: We compute the leading coefficient and Taylor expansion of the `L`-series of a rank `2` elliptic curve. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L.num_coeffs() 156 sage: L.derivative(1,E.rank()) 1.51863300057685 sage: L.taylor_series(1,4) -1.27685190980159e-23 + (7.23588070754027e-24)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 32-bit -2.72911738151096e-23 + (1.54658247036311e-23)*z + 0.759316500288427*z^2 - 0.430302337583362*z^3 + O(z^4) # 64-bit NUMBER FIELD: We compute with the Dedekind zeta function of a number field. :: sage: x = var('x') sage: K = NumberField(x**4 - x**2 - 1,'a') sage: L = K.zeta_function(algorithm='gp') sage: L.conductor 400 sage: L.num_coeffs() 264 sage: L(2) 1.10398438736918 sage: L.taylor_series(2,3) 1.10398438736918 - 0.215822638498759*z + 0.279836437522536*z^2 + O(z^3) RAMANUJAN DELTA L-FUNCTION: The coefficients are given by Ramanujan's tau function:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) We redefine the default bound on the coefficients: Deligne's estimate on tau(n) is better than the default coefgrow(n)=`(4n)^{11/2}` (by a factor 1024), so re-defining coefgrow() improves efficiency (slightly faster). :: sage: L.num_coeffs() 12 sage: L.set_coeff_growth('2*n^(11/2)') sage: L.num_coeffs() 11 Now we're ready to evaluate, etc. :: sage: L(1) 0.0374412812685155 sage: L(1, 1.1) 0.0374412812685155 sage: L.taylor_series(1,3) 0.0374412812685155 + 0.0709221123619322*z + 0.0380744761270520*z^2 + O(z^3) """ __gp = None __globals = set() # set of global variables defined in a run of the # computel.gp script that are replaced by indexed copies # in the computel.gp.template __globals_re = None __instance = 0 # Monotonically increasing unique instance ID __n_instances = 0 # Number of currently allocated instances __template_filename = os.path.join(SAGE_EXTCODE, 'pari', 'dokchitser', 'computel.gp.template') __init = False def __new__(cls, *args, **kwargs): inst = super(Dokchitser, cls).__new__(cls, *args, **kwargs) inst.__instance = cls.__instance cls.__n_instances += 1 cls.__instance += 1 return inst def __init__(self, conductor, gammaV, weight, eps, poles=None, residues='automatic', prec=53, init=None): """ Initialization of Dokchitser calculator EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 """ self.conductor = conductor self.gammaV = gammaV self.weight = weight self.eps = eps self.poles = poles if poles is not None else [] self.residues = residues self.prec = prec self.__CC = ComplexField(self.prec) self.__RR = self.__CC._real_field() self.__initialized = False if init is not None: self.init_coeffs(init) def __reduce__(self): D = copy.copy(self.__dict__) if '_Dokchitser__gp' in D: del D['_Dokchitser__gp'] return reduce_load_dokchitser, (D, ) def _repr_(self): return "Dokchitser L-series of conductor %s and weight %s" % ( self.conductor, self.weight) def __del__(self): self._teardown_gp(self.__instance) def gp(self): """ Return the gp interpreter that is used to implement this Dokchitser L-function. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L(2) 0.546048036215014 sage: L.gp() PARI/GP interpreter """ if self.__gp is None: self._instantiate_gp() elif self.__initialized: return self.__gp self.__initialized = True with open(self.__template_filename) as tf: template = string.Template(tf.read()) tmp_script = os.path.join(SAGE_TMP, 'computel_%s.gp' % self.__instance) with open(tmp_script, 'w') as f: f.write(template.substitute(i=str(self.__instance))) try: self.__gp.read(tmp_script) finally: os.unlink(tmp_script) self._gp_eval('default(realprecision, %s)' % (self.prec // 3 + 2)) self._gp_set_inst('conductor', self.conductor) self._gp_set_inst('gammaV', self.gammaV) self._gp_set_inst('weight', self.weight) self._gp_set_inst('sgn', self.eps) self._gp_set_inst('Lpoles', self.poles) self._gp_set_inst('Lresidues', self.residues) return self.__gp @classmethod def _instantiate_gp(cls): from sage.env import DOT_SAGE logfile = os.path.join(DOT_SAGE, 'dokchitser.log') cls.__gp = sage.interfaces.gp.Gp(script_subdirectory='dokchitser', logfile=logfile) # Read the script template and parse out all indexed global variables # (easy because they all end in "_$i" and there's nothing else in the # script that uses $) global_re = re.compile(r'([a-zA-Z0-9]+)_\$i') with open(cls.__template_filename) as f: for line in f: for m in global_re.finditer(line): cls.__globals.add(m.group(1)) cls.__globals_re = re.compile( '([^a-zA-Z0-9_]|^)(%s)([^a-zA-Z0-9_]|$)' % '|'.join(cls.__globals)) return @classmethod def _teardown_gp(cls, instance=None): cls.__n_instances -= 1 if cls.__n_instances == 0: cls.__gp.quit() elif instance is not None: # Clean up all global variables created by this instance for varname in cls.__globals: cls.__gp.eval('kill(%s_%s)' % (varname, instance)) def _gp_call_inst(self, func, *args): """ Call the specified PARI function in the GP interpreter, with the instance number of this `Dokchitser` instance automatically appended. For example, ``self._gp_call_inst('L', 1)`` is equivalent to ``self.gp().eval('L_N(1)')`` where ``N`` is ``self.__instance``. """ cmd = '%s_%d(%s)' % (func, self.__instance, ','.join( str(a) for a in args)) return self._gp_eval(cmd) def _gp_set_inst(self, varname, value): """ Like ``_gp_call_inst`` but sets the variable given by ``varname`` to the given value, appending ``_N`` to the variable name. If ``varname`` is a function (e.g. ``'func(n)'``) then this sets ``func_N(n)``). """ if '(' in varname: funcname, args = varname.split('(', 1) varname = '%s_%s(%s' % (funcname, self.__instance, args) else: varname = '%s_%s' % (varname, self.__instance) cmd = '%s = %s' % (varname, value) return self._gp_eval(cmd) def _gp_eval(self, s): try: t = self.gp().eval(s) except (RuntimeError, TypeError): raise RuntimeError( "Unable to create L-series, due to precision or other limits in PARI." ) if not self.__init and '***' in t: # After init_coeffs is called, future calls to this method should # return the full output for further parsing raise RuntimeError( "Unable to create L-series, due to precision or other limits in PARI." ) return t def __check_init(self): if not self.__init: raise ValueError( "you must call init_coeffs on the L-function first") def num_coeffs(self, T=1): """ Return number of coefficients `a_n` that are needed in order to perform most relevant `L`-function computations to the desired precision. EXAMPLES:: sage: E = EllipticCurve('11a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L.num_coeffs() 26 sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L.num_coeffs() 568 sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.num_coeffs() 4 Verify that ``num_coeffs`` works with non-real spectral parameters, e.g. for the L-function of the level 10 Maass form with eigenvalue 2.7341055592527126:: sage: ev = 2.7341055592527126 sage: L = Dokchitser(conductor=10, gammaV=[ev*i, -ev*i],weight=2,eps=1) sage: L.num_coeffs() 26 """ return Integer(self._gp_call_inst('cflength', T)) def init_coeffs(self, v, cutoff=1, w=None, pari_precode='', max_imaginary_part=0, max_asymp_coeffs=40): """ Set the coefficients `a_n` of the `L`-series. If `L(s)` is not equal to its dual, pass the coefficients of the dual as the second optional argument. INPUT: - ``v`` -- list of complex numbers or string (pari function of k) - ``cutoff`` -- real number = 1 (default: 1) - ``w`` -- list of complex numbers or string (pari function of k) - ``pari_precode`` -- some code to execute in pari before calling initLdata - ``max_imaginary_part`` -- (default: 0): redefine if you want to compute L(s) for s having large imaginary part, - ``max_asymp_coeffs`` -- (default: 40): at most this many terms are generated in asymptotic series for phi(t) and G(s,t). EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) Evaluate the resulting L-function at a point, and compare with the answer that one gets "by definition" (of L-function attached to a modular form):: sage: L(14) 0.998583063162746 sage: a = delta_qexp(1000) sage: sum(a[n]/float(n)^14 for n in range(1,1000)) 0.9985830631627459 Illustrate that one can give a list of complex numbers for v (see :trac:`10937`):: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:]) sage: L2(14) 0.998583063162746 TESTS: Verify that setting the `w` parameter does not raise an error (see :trac:`10937`). Note that the meaning of `w` does not seem to be documented anywhere in Dokchitser's package yet, so there is no claim that the example below is meaningful! :: sage: L2 = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: L2.init_coeffs(list(delta_qexp(1000))[1:], w=[1..1000]) """ if isinstance(v, tuple) and w is None: v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs = v if pari_precode: # Must have __globals_re instantiated if self.__gp is None: self._instantiate_gp() def repl(m): return '%s%s_%d%s' % (m.group(1), m.group(2), self.__instance, m.group(3)) # If any of the pre-code contains references to some of the # templated global variables we must replace those as well pari_precode = self.__globals_re.sub(repl, pari_precode) if pari_precode != '': self._gp_eval(pari_precode) RR = self.__CC._real_field() cutoff = RR(cutoff) if isinstance(v, str): if w is None: self._gp_call_inst('initLdata', '"%s"' % v, cutoff) else: self._gp_call_inst('initLdata', '"%s"' % v, cutoff, '"%s"' % w) elif not isinstance(v, (list, tuple)): raise TypeError("v (=%s) must be a list, tuple, or string" % v) else: CC = self.__CC v = ','.join([CC(a)._pari_init_() for a in v]) self._gp_eval('Avec = [%s]' % v) if w is None: self._gp_call_inst('initLdata', '"Avec[k]"', cutoff) else: w = ','.join([CC(a)._pari_init_() for a in w]) self._gp_eval('Bvec = [%s]' % w) self._gp_call_inst('initLdata', '"Avec[k]"', cutoff, '"Bvec[k]"') self.__init = (v, cutoff, w, pari_precode, max_imaginary_part, max_asymp_coeffs) def __to_CC(self, s): s = s.replace('.E', '.0E').replace(' ', '') return self.__CC(sage_eval(s, locals={'I': self.__CC.gen(0)})) def _clear_value_cache(self): del self.__values def __call__(self, s, c=None): r""" INPUT: - ``s`` -- complex number .. NOTE:: Evaluation of the function takes a long time, so each evaluation is cached. Call ``self._clear_value_cache()`` to clear the evaluation cache. EXAMPLES:: sage: E = EllipticCurve('5077a') sage: L = E.lseries().dokchitser(100, algorithm='gp') sage: L(1) 0.00000000000000000000000000000 sage: L(1+I) -1.3085436607849493358323930438 + 0.81298000036784359634835412129*I """ self.__check_init() s = self.__CC(s) try: return self.__values[s] except AttributeError: self.__values = {} except KeyError: pass z = self._gp_call_inst('L', s) if 'pole' in z: print(z) raise ArithmeticError elif '***' in z: print(z) raise RuntimeError elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits', 'decimal digits') verbose(msg, level=-1) ans = self.__to_CC(z[i + 1:]) self.__values[s] = ans return ans ans = self.__to_CC(z) self.__values[s] = ans return ans def derivative(self, s, k=1): r""" Return the `k`-th derivative of the `L`-series at `s`. .. WARNING:: If `k` is greater than the order of vanishing of `L` at `s` you may get nonsense. EXAMPLES:: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L.derivative(1,E.rank()) 1.51863300057685 """ self.__check_init() s = self.__CC(s) k = Integer(k) z = self._gp_call_inst('L', s, '', k) if 'pole' in z: raise ArithmeticError(z) elif 'Warning' in z: i = z.rfind('\n') msg = z[:i].replace('digits', 'decimal digits') verbose(msg, level=-1) return self.__CC(z[i:]) return self.__CC(z) def taylor_series(self, a=0, k=6, var='z'): r""" Return the first `k` terms of the Taylor series expansion of the `L`-series about `a`. This is returned as a series in ``var``, where you should view ``var`` as equal to `s-a`. Thus this function returns the formal power series whose coefficients are `L^{(n)}(a)/n!`. INPUT: - ``a`` -- complex number (default: 0); point about which to expand - ``k`` -- integer (default: 6), series is `O(``var``^k)` - ``var`` -- string (default: 'z'), variable of power series EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.taylor_series(2, 3) 1.64493406684823 - 0.937548254315844*z + 0.994640117149451*z^2 + O(z^3) sage: E = EllipticCurve('37a') sage: L = E.lseries().dokchitser(algorithm='gp') sage: L.taylor_series(1) 0.000000000000000 + 0.305999773834052*z + 0.186547797268162*z^2 - 0.136791463097188*z^3 + 0.0161066468496401*z^4 + 0.0185955175398802*z^5 + O(z^6) We compute a Taylor series where each coefficient is to high precision. :: sage: E = EllipticCurve('389a') sage: L = E.lseries().dokchitser(200, algorithm='gp') sage: L.taylor_series(1,3) ...e-82 + (...e-82)*z + 0.75931650028842677023019260789472201907809751649492435158581*z^2 + O(z^3) Check that :trac:`25402` is fixed:: sage: L = EllipticCurve("24a1").modular_form().lseries() sage: L.taylor_series(-1, 3) 0.000000000000000 - 0.702565506265199*z + 0.638929001045535*z^2 + O(z^3) Check that :trac:`25965` is fixed:: sage: L2 = EllipticCurve("37a1").modular_form().lseries(); L2 L-series associated to the cusp form q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + O(q^6) sage: L2.taylor_series(0,4) 0.000000000000000 - 0.357620466127498*z + 0.273373112603865*z^2 + 0.303362857047671*z^3 + O(z^4) sage: L2.taylor_series(0,1) O(z^1) sage: L2(0) 0.000000000000000 """ self.__check_init() a = self.__CC(a) k = Integer(k) try: z = self._gp_call_inst('Lseries', a, '', k - 1) z = self.gp()('Vecrev(Pol(%s))' % z) except TypeError as msg: raise RuntimeError( "%s\nUnable to compute Taylor expansion (try lowering the number of terms)" % msg) r = repr(z) if 'pole' in r: raise ArithmeticError(r) elif 'Warning' in r: i = r.rfind('\n') msg = r[:i].replace('digits', 'decimal digits') verbose(msg, level=-1) v = list(z) K = self.__CC v = [K(repr(x)) for x in v] R = self.__CC[[var]] return R(v, k) def check_functional_equation(self, T=1.2): r""" Verifies how well numerically the functional equation is satisfied, and also determines the residues if ``self.poles != []`` and residues='automatic'. More specifically: for `T>1` (default 1.2), ``self.check_functional_equation(T)`` should ideally return 0 (to the current precision). - if what this function returns does not look like 0 at all, probably the functional equation is wrong (i.e. some of the parameters gammaV, conductor etc., or the coefficients are wrong), - if checkfeq(T) is to be used, more coefficients have to be generated (approximately T times more), e.g. call cflength(1.3), initLdata("a(k)",1.3), checkfeq(1.3) - T=1 always (!) returns 0, so T has to be away from 1 - default value `T=1.2` seems to give a reasonable balance - if you don't have to verify the functional equation or the L-values, call num_coeffs(1) and initLdata("a(k)",1), you need slightly less coefficients. EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=1, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() # abs tol 1e-19 -2.71050543121376e-20 If we choose the sign in functional equation for the `\zeta` function incorrectly, the functional equation doesn't check out. :: sage: L = Dokchitser(conductor=1, gammaV=[0], weight=1, eps=-11, poles=[1], residues=[-1], init='1') sage: L.check_functional_equation() -9.73967861488124 """ self.__check_init() z = self._gp_call_inst('checkfeq', T) z = z.replace(' ', '') return self.__CC(z) def set_coeff_growth(self, coefgrow): r""" You might have to redefine the coefficient growth function if the `a_n` of the `L`-series are not given by the following PARI function:: coefgrow(n) = if(length(Lpoles), 1.5*n^(vecmax(real(Lpoles))-1), sqrt(4*n)^(weight-1)); INPUT: - ``coefgrow`` -- string that evaluates to a PARI function of n that defines a coefgrow function. EXAMPLES:: sage: L = Dokchitser(conductor=1, gammaV=[0,1], weight=12, eps=1) sage: pari_precode = 'tau(n)=(5*sigma(n,3)+7*sigma(n,5))*n/12 - 35*sum(k=1,n-1,(6*k-4*(n-k))*sigma(k,3)*sigma(n-k,5))' sage: L.init_coeffs('tau(k)', pari_precode=pari_precode) sage: L.set_coeff_growth('2*n^(11/2)') sage: L(1) 0.0374412812685155 """ if not isinstance(coefgrow, str): raise TypeError("coefgrow must be a string") self._gp_set_inst('coefgrow(n)', coefgrow.replace('\n', ' '))