Would anyone know a reference that proves the basic facts about Artin $L$-functions at $\Re(s) = 1$? Namely, the non-vanishing and holomorphicity for non-trivial characters.
I assume this was done in Artin's original paper, but a modern source would be most welcome.
I think nonvanishing and meromorphicity follow from Artin's theorem on induced characters, class field theory, and the analogous results for $L$-functions of Hecke characters. In fact, if I recall correctly, this was the motivation for Artin's theorem.
I also had a hard time finding a reference for this a while back.
See page 225 of Chapter VIII "Zeta-Functions and L-functions" by Heilbronn in "Algebraic number theory" edited by Cassels and Fröhlich, MR0218327
Also see $\S$12 of Chapter XIII of Weil's book "Basic Number Theory", MR0234930
I don't know of a text that comes out and says this directly, but it can be gleaned by combining a few key results. I summarize below.
As I understand matters, we can only prove such results for Artin $L$-function associated to a Galois extension $L/K$, where $L$ and $K$ are number fields with absolute discriminants $D_L$ and $D_K$. If $F\in\{L,K\}$, then one can prove a zero-free region of the form
$\mathrm{Re}(s)\geq 1-\frac{c}{\log(D_F(3+|\mathrm{Im}(s)|)^{[F:\mathbb{Q}]})}$
for the Dedekind zeta function $\zeta_F(s)$, apart from a real exceptional Landau-Siegel zero. Thus, apart from a Landau-Siegel zero of $\zeta_L(s)$, the ratio $\zeta_L(s)/\zeta_K(s)$ is holomorphic and nonzero in the region
$\mathrm{Re}(s)\geq 1-\frac{c}{\log(D_L(3+|\mathrm{Im}(s)|)^{[F:\mathbb{Q}]})}$.
(This is proved in several standard analytic number theory sources. The most natural source is Lagarias-Odlyzko, but it's hard to find a copy. Iwaniec and Kowalski prove a weaker form of this, which suffices if all you care about is behavior on $\mathrm{Re}(s)=1$.) Artin proved that
$\zeta_L(s)/\zeta_K(s) = \prod_{\rho} L(s,\rho)^{\dim(\rho)}$,
where $\rho$ ranges over the irreducible Artin representations of $\mathrm{Gal}(L/K)$. Brauer (Annals, 1948?) proved that as $H$ ranges over the elementary subgroups of $G$ and $\chi$ ranges over the (Hecke) characters in the dual group of $H$, there exist integers $c_{\chi,\rho}$ such that
$L(s,\rho) = \prod_{H\subseteq G}\prod_{\chi \in \widehat{H}} L(s,\chi)^{c_{\chi,\rho}}$.
(These are classical, and the original sources are likely to be in German. You could glean these from Neukirch's book on algebraic number theory.). Hecke proved that each such $L(s,\chi)$ is holomorphic and nonzero in the same region as $\zeta_L(s)/\zeta_K(s)$. (Probably the proof that is easiest to access is in Murty and Petersen (Transactions of the AMS, 2013).)
Piecing all of this together, we have a region containing the line $\mathrm{Re}(s)=1$ in which the Artin $L$-functions associated to the extension $L/K$ are holomorphic and nonzero.
Concretely from Artin's theorem on induced characters we get that for a representation $\rho$ of $Gal(L/K)$ $$L(s,\rho)=\prod_j L(s,\chi_j)^{c_j}$$ where the $c_j\in \Bbb{Q}$ and the $\chi_j$ are 1-dimensional characters of subgroups $Gal(L/F_j)$ of $Gal(L/K)$.
The hard part:
By class field theory $L(s,\chi_j)=L(s,\psi_j)$ where $\psi_j$ is a finite order Hecke character of $F_j$. By say Tate's thesis then $L(s,\psi_j)$ is entire for $\chi_j$ non-trivial, when it is trivial then $L(s,\psi_j)=\zeta_{F_j}(s)$ and $(s-1)\zeta_{F_j}(s)$ is entire and non-zero at $s=1$.
$\prod_{l=1}^{order(\chi_j)} L(s,\chi_j^l)=\zeta_{E_j}(s)$ for some abelian extension $E_j/F_j$.
$\zeta_{E_j}(s)$ doesn't vanish on $\Re(s)=1$ (can be improved to a zero-free region, effective once $L$ is fixed) from the same proof as for $\zeta(s)$, using that it is analytic away from a simple pole at $s=1$ and $\log \zeta_{E_j}(s)$ has non-negative coefficients. Whence $L(s,\chi_j)$ doesn't vanish on $\Re(s)=1$, and $L(s,\rho)$ is analytic non-zero on $\Re(s)=1,s\ne 1$.
The behavior at $s=1$ is found by writing the $L(s,\chi_j)$ as Artin L-functions $L(s,\rho_j)$ with $\rho_j$ a representation of $Gal(L/K)$,
$\frac1{|G|}\sum_{g\in Gal(L/K)} tr(\rho_j(g))=1$ if $\chi_j$ is trivial and $=0$ otherwise,
giving that the $r\in \Bbb{Q}$ such that $(s-1)^r L(s,\rho)$ is analytic non-zero at $s=1$ is $\frac1{|G|}\sum_{g\in Gal(L/K)} tr(\rho(g))$, which is an integer $\ge 0$.
