Let $E/F$ be a finite Galois extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $F$ containing the infinite places. For $\chi$ an irreducible complex character of $G$, let $L_S(s,\chi)$ denote the $S$-truncated Artin $L$-function attached to $\chi$. If $1_G$ denotes the trivial character then $L_S(s,1_G)$ becomes the $S$-truncated Dedekind-zeta function of $F$, and so by the analytic class number formula this has a simple pole at $s=1$. However, if $\chi$ is a non-trivial irreducible character then $L_S(s,\chi)$ has neither a zero nor a pole at $s=1$. One can prove this by reducing to linear characters using Brauer induction; the case of linear characters comes down to the same claim for Hecke $L$-series, which is proven in Lemma 13.3 of *Cohomology of number fields* (available here).

Another way of saying all this is that if $\chi$ is a virtual complex character of $G$ then $L_S(s,\chi)$ has a pole of order $\langle \chi, 1_G \rangle_G$ (the usual inner product of characters of $G$) at $s=1$.

My question is: can you provide a neat reference for this fact that I can quickly cite in a paper I am working on without having to go into the explanation above?