def zeros_in_interval(self, x, y, stepsize):
r"""
Return the imaginary parts of (most of) the nontrivial zeros
on the critical line `\Re(s)=1` with positive imaginary part
between ``x`` and ``y``, along with a technical quantity for each.
INPUT:
- ``x``-- positive floating point number
- ``y``-- positive floating point number
- ``stepsize`` -- positive floating point number
OUTPUT:
- list of pairs ``(zero, S(T))``.
Rubinstein writes: The first column outputs the imaginary part
of the zero, the second column a quantity related to ``S(T)`` (it
increases roughly by 2 whenever a sign change, i.e. pair of
zeros, is missed). Higher up the critical strip you should use
a smaller stepsize so as not to miss zeros.
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: E.lseries().zeros_in_interval(6, 10, 0.1) # long time
[(6.87039122, 0.248922780), (8.01433081, -0.140168533), (9.93309835, -0.129943029)]
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.zeros_in_interval(x, y, stepsize, L=self.__E)
def zeros_in_interval(self, x, y, stepsize):
r"""
Return the imaginary parts of (most of) the nontrivial zeros
on the critical line $\Re(s)=1$ with positive imaginary part
between $x$ and $y$, along with a technical quantity for each.
INPUT:
x, y, stepsize -- positive floating point numbers
OUTPUT:
list of pairs (zero, S(T)).
Rubinstein writes: The first column outputs the imaginary part
of the zero, the second column a quantity related to S(T) (it
increases roughly by 2 whenever a sign change, i.e. pair of
zeros, is missed). Higher up the critical strip you should use
a smaller stepsize so as not to miss zeros.
EXAMPLES::
sage: E = EllipticCurve('37a')
sage: E.lseries().zeros_in_interval(6, 10, 0.1) # long time
[(6.87039122, 0.248922780), (8.01433081, -0.140168533), (9.93309835, -0.129943029)]
"""
from sage.lfunctions.lcalc import lcalc
return lcalc.zeros_in_interval(x, y, stepsize, L=self.__E)