def taylor_series(self, s, k=6, var='z'):
r"""
Return the first `k` terms of the Taylor series expansion
of the `L`-series about `s`.
This is returned as a formal power series in ``var``.
INPUT:
- ``s`` -- complex number; point about which to expand
- ``k`` -- optional integer (default: 6), series is
`O(``var``^k)`
- ``var`` -- optional string (default: 'z'), variable of power
series
EXAMPLES::
sage: from sage.lfunctions.pari import lfun_number_field, LFunction
sage: lf = lfun_number_field(QQ)
sage: L = LFunction(lf)
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="pari")
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="pari")
sage: L.taylor_series(1,3)
2...e-63 + (...e-63)*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)
"""
pt = pari.Ser([s, 1], d=k) # s + x + O(x^k)
B = PowerSeriesRing(self._CC, var)
return B(pari.lfun(self._L, pt, precision=self.prec))
def derivative(self, s, D=1):
"""
Return the derivative of the L-function at point s and order D.
INPUT:
- ``s`` -- complex number
- ``D`` -- optional integer (default 1)
EXAMPLES::
sage: from sage.lfunctions.pari import lfun_number_field, LFunction
sage: L = LFunction(lfun_number_field(QQ))
sage: L.derivative(2)
-0.937548254315844
"""
s = self._CCin(s)
R = self._CC
return R(pari.lfun(self._L, s, D, precision=self.prec))
def __call__(self, s):
r"""
Return the value of the L-function at point ``s``.
INPUT:
- ``s`` -- complex number
.. NOTE::
Evaluation of the function takes a long time, so each
evaluation is cached. Call :meth:`_clear_value_cache` to
clear the evaluation cache.
EXAMPLES::
sage: E = EllipticCurve('5077a')
sage: L = E.lseries().dokchitser(100, algorithm="pari")
sage: L(1)
0.00000000000000000000000000000
sage: L(1+I)
-1.3085436607849493358323930438 + 0.81298000036784359634835412129*I
"""
s = self._CC(s)
try:
return self.__values[s]
except AttributeError:
self.__values = {}
except KeyError:
pass
try:
value = self._CC(pari.lfun(self._L, s, precision=self.prec))
except NameError:
raise ArithmeticError('pole here')
else:
return value
