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.