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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.

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    $\begingroup$ 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. $\endgroup$
    – reuns
    Commented May 18, 2020 at 4:14

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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 $1$.

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