derivatives of Artin L-functions

2013-01-22T17:34:23Z

This is a vague question: I'm sorry for that.

Let's start with $\chi$ a (primitive odd) Dirichlet character modulo $n$ and look at the corresponding L-function $$ L(s, \chi)=\sum \frac{\chi(n)}{n^s}. $$ A classical formula due to Hurwitz computes the logarithmic derivative of $L(s, \chi)$ at $s=0$ in terms of the Gamma function: $$ \frac{L'(0, \chi)}{L(0,\chi)}=-\log n + \frac{\sum \log \Gamma(1-\frac{u}{n})\chi(u)}{\sum \chi(u) \frac{u}{n}} $$ (By the way, does anybody know of some good references for this formula?)

That's the abelian case. Now imagine $L/\mathbb{Q}$ is a Galois field extension with non-abelian Galois group $G$, and let $\chi$ be a character of $G$. Now one has the Artin L-function.

Question: Is there any similar formula know for $\frac{L'(0, \chi)}{L(0, \chi)}$?

My naive guess would be the following: by Brauer's theorem one knows that $\chi$ can be written as a sum with rational coefficients of characters induced by cyclic subgroups of $G$, so $\log L(s, \chi)$ should be a sum with rational coefficients of some Dirichlet L-functions and for each one there is a such a formula. However, I was unable to find something like that in the literature. Is there some mistake here? If rational coefficients create some trouble, one can also write $\chi$ as a combination with integer coefficients of characters induced from elementary subgroups. Does it help?

Another question: is there any relation between Jacobi sums and these derivatives?

Thanks for your help.

Answer by magnilor for derivatives of Artin L-functions

2013-01-23T14:35:54Z

François is absolutely right. In my question I was assuming that the logarithmic derivative exists. So assume the condition he worked out is satisfied. What can be said about the values of this derivatives in terms of Gamma values or Jacobi sums?

derivatives of Artin L-functions - MathOverflow

derivatives of Artin L-functions

Answer by magnilor for derivatives of Artin L-functions