I know that for $\chi\not =1$ (i.e. Dirichlet character of modulo $m$) $L(1,\chi)\not= 0$.
In Serre's book (A Course in Arithmetic page 74) he gives 2 definitions of the complex logarithm of $Log(L(1,\chi))$:
For the second definition, I believe he is using the fact that $L(1,\chi)$ is analytic and non-zero in a region around 1 so it has a an analytic function $g$ s.t $e^g=f$ in this small region.
But how do I use this g to get a series definition of $Log(L(s,\chi))$?
I don't quite understand the first definition. Is he composing $L(s,\chi)$ with some branch of the complex logarithm? If so, how can we confirm that the image of $L(s,\chi)$ falls with the domain of that branch?
Can someone care to explain how what Serre means by $Log(L(s,\chi))$?
With the branch of $\log $ analytic for $|z-1|<1$ and $\log(1)=0$ and $\chi(n)$ completely multiplicative periodic (so that $|\chi(n)|=0$ or $1$) then $$F(s)=-\sum_p \log (1-\chi(p)p^{-s})$$ converges and is analytic for $\Re(s) > 1$ and we have $$\exp(F(s))=\prod_p \frac1{1-\chi(p)p^{-s}}=\sum_{n\ge 1}\chi(n)n^{-s}$$ Now for other kind of functions you should see that $\log G(s)$ is obtained by integrating $\frac{G'(s)}{G(s)}$ which is meromorphic if $G$ is meromorphic. Then the branches of $\log G(s)$ are found from the closed-curves such that $\int_\gamma \frac{G'(s)}{G(s)}ds$ is zero or non-zero. In particular if $G$ is analytic non-zero on a simply connected open then $\log G(s)$ extends analytically to it.