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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?

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    $\begingroup$ See Heilbronn's article "Zeta functions and L-functions" in the book "Algebraic number Theory" by Cassels and Frohlich. On the last line of the remarks following Theorem 7 on page 225, Heilbronn writes "Artin L-functions formed with non-principal characters are, in addition, regular and non-zero for $\sigma \geq 1$". This is a reference, but as a reader, I would prefer to see the proof you have skethced. $\endgroup$ – Venkataramana Aug 3 '16 at 11:07
  • $\begingroup$ @ Venkataramana - thanks for this. I suppose that a clear reference that also gives a (sketch) proof would be ideal. $\endgroup$ – Henri Johnston Aug 3 '16 at 11:13
  • $\begingroup$ not at all! [I do not know how to correct my typo in comments: "sketched" in place of "skethced"]. Maybe Brauer's paper (cited in Heilbronn's article) contains the proof of the statement. $\endgroup$ – Venkataramana Aug 3 '16 at 12:47
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    $\begingroup$ Just going off memory here, but it seems likely to be in a paper of Martinet where he gives a long intro to Artin L-functions, Serre's Modular forms of weight one and Galois representations paper, and/or Serre's Sem Bourbaki on Brauer's character theorem. $\endgroup$ – Kimball Aug 4 '16 at 10:51
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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.

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