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. 



Logarithms of $L$-functions of irreducible characters of Galois group
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 $1$.


