How to define L-series?

[Adrián Barquero]

Dear All,

I was trying to define Dirichlet L-functions in SAGE, and while looking online for how to actually do it, I found the following tutorial from SAGE Days 33

http://wiki.sagemath.org/days33/lfunction/tutorial

In that tutorial it is described how to define different types of L-series, for example Dedekind zeta functions, Dirichlet L-functions and L-functions of Elliptic Curves by using the method LSeries().

Nevertheless, when I try to run the method, for example to define the Dirichlet L-function attached to some Dirichlet character, I get the error message

NameError: name 'LSeries' is not defined

Thus, I would like to ask if you can help me identify what it is that I'm doing wrong, or how to define Dirichlet L-functions in SAGE.

PS: The version of SAGE that I have installed in my computer is Version 5.3-OSX-64bit-10.6  (5.3-OSX-64bit-10.6), but I also tried running the same code online and it didn't work either.

Thank you very much for your help.

[John Cremona]

I think that the code described in that tutorial was under development
at that Sage Days, but has never been completed. I hope that someone
who worked on it can give more information, but certainly even in Sage
version 5.10 (the latest release) search_src("LSeries") reveals
nothing relevant.

John Cremona

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[kcrisman]

Just FYI, I asked http://ask.sagemath.org/question/2540/defining-dirichlet-series a while ago and have wanted to know the answer to a very similar question for several years :) in case someone does know how to do it.

[Georgi Guninski]

Currently you need to load "eulerprod.py" (search for it), maybe from here:

http://purplesage.googlecode.com/hg-history/6bac823fb2ef13c9c602203591587049acfe3863/psage/lseries/eulerprod.py

This works for me:

sage: load eulerprod.py
sage: Z1.<Z>=ZZ[]
sage: Nf.<v>=NumberField(Z**2+1)
sage: L=LSeries(Nf)
sage: L
Dedekind Zeta function of Number Field in v with defining polynomial Z^2 + 1
sage: L(1/2+I)
0.302675790657936779964948103095684788837343183147516784341448029748974263532441111279452499 - 0.517840900734412719155534258211294840216830050895051536044014429256591893818783220651514849*I

[Michael Beeson]

I tried to follow Guninski's instructions.   I found eulerprod.py and put it in the right directory 

and then I got an error due to not having psage.ellcurve.lseries.helper.

I googled for that file and put it in helper.py but I still get an error.  Maybe the file 

needs to have a complicated path matching the module name?  What do I do now?

sage: load eulerprod.py

---------------------------------------------------------------------------

ImportError                               Traceback (most recent call last)

/Users/beeson/<ipython console> in <module>()

/Applications/sage/local/lib/python2.6/site-packages/sage/misc/preparser.pyc in load(filename, globals, attach)

   1634 

   1635     if fpath.endswith('.py'):

-> 1636         execfile(fpath, globals)

   1637     elif fpath.endswith('.sage'):

   1638         if (attach and attach_debug_mode) or ((not attach) and load_debug_mode):

/Users/beeson/eulerprod.py in <module>()

      4 from sage.rings.all import is_RationalField, ZZ, Integer, QQ, O, ComplexField, CDF, primes, infinity as oo

      5 from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve

----> 6 from psage.ellcurve.lseries.helper import extend_multiplicatively_generic

      7 from sage.misc.all import prod

      8 from sage.modular.abvar.abvar import is_ModularAbelianVariety

ImportError: No module named psage.ellcurve.lseries.helper