def twist_values(self, s, dmin, dmax):
r"""
Return values of $L(E, s, \chi_d)$ for each quadratic
character $\chi_d$ for $d_{\min} \leq d \leq d_{\max}$.
\note{The L-series is normalized so that the center of the
critical strip is 1.}
INPUT:
- ``s`` -- complex numbers
- ``dmin`` -- integer
- ``dmax`` -- integer
OUTPUT:
list of pairs (d, L(E, s,chi_d))
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: vals = E.lseries().twist_values(1, -12, -4)
sage: vals # abs tol 1e-17
[(-11, 1.47824342), (-8, 8.9590946e-18), (-7, 1.85307619), (-4, 2.45138938)]
sage: F = E.quadratic_twist(-8)
sage: F.rank()
1
sage: F = E.quadratic_twist(-7)
sage: F.rank()
0
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.twist_values(s - RationalField()('1/2'), dmin, dmax, L=self.__E)
def twist_values(self, s, dmin, dmax):
r"""
Return values of $L(E, s, \chi_d)$ for each quadratic
character $\chi_d$ for $d_{\min} \leq d \leq d_{\max}$.
\note{The L-series is normalized so that the center of the
critical strip is 1.}
INPUT:
s -- complex numbers
dmin -- integer
dmax -- integer
OUTPUT:
list -- list of pairs (d, L(E, s,chi_d))
EXAMPLES:
sage: E = EllipticCurve('37a')
sage: E.lseries().twist_values(1, -12, -4) # slightly random output depending on architecture
[(-11, 1.4782434171), (-8, 0), (-7, 1.8530761916), (-4, 2.4513893817)]
sage: F = E.quadratic_twist(-8)
sage: F.rank()
1
sage: F = E.quadratic_twist(-7)
sage: F.rank()
0
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.twist_values(s - RationalField()('1/2'), dmin, dmax, L=self.__E)
def twist_values(self, s, dmin, dmax):
r"""
Return values of $L(E, s, \chi_d)$ for each quadratic
character $\chi_d$ for $d_{\min} \leq d \leq d_{\max}$.
\note{The L-series is normalized so that the center of the
critical strip is 1.}
INPUT:
s -- complex numbers
dmin -- integer
dmax -- integer
OUTPUT:
list -- list of pairs (d, L(E, s,chi_d))
EXAMPLES:
sage: E = EllipticCurve('37a')
sage: E.lseries().twist_values(1, -12, -4) # slightly random output depending on architecture
[(-11, 1.4782434171), (-8, 0), (-7, 1.8530761916), (-4, 2.4513893817)]
sage: F = E.quadratic_twist(-8)
sage: F.rank()
1
sage: F = E.quadratic_twist(-7)
sage: F.rank()
0
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.twist_values(s - RationalField()('1/2'),
dmin,
dmax,
L=self.__E)
def twist_values(self, s, dmin, dmax):
r"""
Return values of `L(E, s, \chi_d)` for each quadratic
character `\chi_d` for `d_{\min} \leq d \leq d_{\max}`.
.. note::
The L-series is normalized so that the center of the
critical strip is 1.
INPUT:
- ``s`` -- complex numbers
- ``dmin`` -- integer
- ``dmax`` -- integer
OUTPUT:
- list of pairs `(d, L(E, s, \chi_d))`
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: vals = E.lseries().twist_values(1, -12, -4)
sage: vals[0][0]
-11
sage: vals[0][1] # abs tol 1e-8
1.47824342 + 0.0*I
sage: vals[1][0]
-8
sage: vals[1][1] # abs tol 1e-8
0.0 + 0.0*I
sage: vals[2][0]
-7
sage: vals[2][1] # abs tol 1e-8
1.85307619 + 0.0*I
sage: vals[3][0]
-4
sage: vals[3][1] # abs tol 1e-8
2.45138938 + 0.0*I
sage: F = E.quadratic_twist(-8)
sage: F.rank()
1
sage: F = E.quadratic_twist(-7)
sage: F.rank()
0
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.twist_values(s - RationalField()('1/2'),
dmin,
dmax,
L=self.__E)
