How to calculate $L'(1,\chi)/L(1,\chi)$?

asked 2012-05-08 10:14:26 -0600

anonymous user

Anonymous

Question as in title, where $L(s,\chi)$ is the Dirichlet $L$-function associated with the nontrivial character modulo $3$. Please provide complete SAGE code. Thank you in advance.

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Comments

Maybe you could start by showing us how much you have already?

niles gravatar imageniles ( 2012-05-08 10:47:22 -0600 )edit

Have you had a look at the [L-function tutorial](http://wiki.sagemath.org/days33/lfunction/tutorial) from Sage days 33? In particular, it has some fairly recent comments on the development status. I have no idea whether Sage yet implements, for example, the insights in Ihara, Murty & Shimura's paper from circa 2007, [On the Logarithmic Derivative of Dirichlet L-Functions at s=1](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/emeritus/ihara/Publications-and-Recent-Preprints/RecentArticles/pdf-files/IMS.main.pdf).

bgins gravatar imagebgins ( 2012-05-08 10:55:22 -0600 )edit

Actually I managed to calculate the particular example "by hand", using that $L(1,\chi)=\sum\chi(n)/n$ and $L'(1,\chi)=-\sum\chi(n)\log(n)/n$. Of course it would be nicer to use some built-in function for that purpose. Note that I am a beginner at SAGE. Also, I looked at http://wiki.sagemath.org/days33/lfunction/tutorial, but the command "LSeries" did not work at http://www.sagenb.org/ (upon calling "L=LSeries(DirichletGroup(3).0)" I get the error message "name 'LSeries' is not defined").

Anonymous gravatar imageAnonymous ( 2012-05-08 11:28:29 -0600 )edit