# Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $$L$$ functions of Dirichlet characters $$\chi$$ of $$(\mathbb Z / m\mathbb Z)^\times$$ satisfy the property that $$\log L(s, \chi)$$ is holomorphic for $$\Re(s) \geq 1$$ if $$\chi$$ is a nontrivial character and if $$\chi$$ is trivial then $$\log L(s, \chi) = \log \frac{1}{s-1}+g(s)$$ for some holomorphic function $$g$$ on $$\Re(s) \geq 1$$.

I wanted to know if the same holds more generally, that is given a finite Galois extension $$K/ \mathbb{Q}$$ with Galois group $$G$$, if $$\chi$$ be an irreducible character of $$G$$, then is it true that $$\log L(s, \chi)$$ is holomorphic for $$\Re(s) \geq 1$$ if $$\chi$$ is a nontrivial character and expressible in the above form for trivial $$\chi$$? I would be really grateful for a proof (or counterexample) or a reference containing either.

• In Neukrich -algebraic number theory it is based on Brauer theorem on induced characters (plus class field theory) giving that $L(s,\rho,L/K)=\prod_j L(s,\psi_j,F_j)^{e_j}$ for some Hecke L-functions of sub-extensions $L/F_j/K$. Add the end of this there is the much simpler Chebotarev method. May 18, 2020 at 4:14

Yes this is the fact that for a non-trivial irreducible Artin character, the associated Artin L-function is holomorphic and non-zero on $$\rm{re}\, s \geq 1$$.
For the trivial character, one just obtains the Riemann zeta function, where there is a pole of order $$1$$.