• LfunctionsAndModularFormsIV
• LcalcExample

Description

lcalcpresentation.sws

This is demo of how to construct a function which creates an lcalc Lfunction object demo.py has sample files.

L-function is assumed to satisfy the following function equation.

\Lambda(s) = \omega \overline{\Lambda(1-\bar s)}

where

\Lambda(s) = Q^s \prod_{j=1}^k {\Gamma(\gamma_j s +\lambda_j)} L(s).

Note that there is a typo in Lfunction_C documentation of the demo worksheet.

Download this file

demo.py

Make sure you have demo.py in the directory from where you invoke the sage command below

Getting into sage environment

pathToSage/sage -sh

Run Ipython

ipython

First couple of lines. The order is important

In [1]: from sage import all as sage

In [2]: import demo

In [3]: E=sage.EllipticCurve([1,1])

In [4]: L=demo.Lfunction_from_elliptic_curve(E)

In [5]: L.find_zeros_via_N(10)
Out[5]: [0.000000000000000, 2.27669243827908, 3.65808391543662, 4.74813565311889, 5.57476376575799, 7.12335236325282, 7.96370743269930, 8.33306194711412, 9.56801749546317, 10.8562725628737]

#Create a sage complex number 1.0 + I and find the value of above L-function at the point
In [6]: L.value( sage.CC(1,1) )
Out[6]: 1.17987989000315 + 0.33328909824226


#
# Below is a demo for Dirichlet L-function
#
In [7]: D=sage.DirichletGroup(15)

In [8]: C=D[3]
In [9]: L2=demo.Lfunction_from_character(C)
Out[9]: L-function with complex Dirichlet coefficients

In [10]: L2.find_zeros_via_N(12)
Out[10]: [2.73460370911883, 5.24301049275449, 8.41468524980490, 10.1876027276453, 11.9073724765466, 13.3373958554156, 15.3661395764978, 17.7723902492167, 18.9769996500883, 20.6276317759696, 21.8839730242089, 23.3384800177599]