Order of vanishing of Artin $L$-functions at $s=1$ 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? 
 A: Iwaniec and Kowalski sketch the argument in their book on analytic number theory, Corollary 5.47 and the discussion preceding it, in pages 142-143.
They do not prove the non-vanishing of Hecke L-functions (but they give the statement, with a zero-free region) or Brauer's induction (but they reference Serre's book on representation theory).

COROLLARY 5.47. Let $\rho$ be a non-trivial irreducible Galois
  representation of $K/\mathbb{Q}$. Then $L(\rho,s)$ has neither poles
  nor zeros on the line $\mathrm{Re}(s)=1$.
Proof. This follows by (5.106) because the L-functions of non-trivial
  Hecke Grossencharakters are entire and do not vanish on the line
  $\mathrm{Re}(s)=1$ (Theorem 5.35).

Equation (5.106) is
$$L(\rho,s)=\prod_i L(\pi_i,s)^{n_i}=\prod_i L(\xi_i,s)^{n_i}$$
where $\mathrm{Tr}\,\rho=\sum n_i \mathrm{Tr} \, \pi_i$, and $\pi_i$ is induced from the abelian character $\xi_i$, and includes a discussion of why it follows from Brauer's induction and invariance of induction.


*

*Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004


On a historical note, Brauer discusses $L(\rho,1)$ in his seminal 1947-1950 papers, but only for abelian characters, attributing the non-vanishing result to Landau. I can't find the result for general $\rho$ in the follow-up papers by van der Wall and Heilbronn, but no doubt Artin was aware of the result, conditional on the induction result.
