Higher-dimensional Artin L-functions

Question

I begin by clarifying that the "higher-dimensional" in my question refers to analogues of Artin L-functions over higher dimensional base schemes than $\mathrm{Spec}(\mathbb{Z})$.

Now for the set-up. This will be very similar to the set-up of Chapter 9 of Serre's book "Lectures on N_X(p)". Let $R$ be a finite type flat $\mathbb{Z}$-algebra of dimension $n$ with fraction field $K$ and let $X = \mathrm{Spec}(R)$. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $Y$ be the normalisation of $X$ in $L$. At any point I am quite happy to shrink $X$. For example, I may assume that $X$ is regular and that $Y/X$ is finite étale.

Now for the analogue of Artin L-functions. For a complex representation $\rho$ of $G$ we define $$L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1}, \quad \text{re}(s) > n.$$ See the above book of Serre for the definition of Frobenius elements in this setting. That the above Euler product converges in the range $\text{re}(s) > n$ follows from the Weil conjectures.

My first question is the following.

1. Have such L-functions been studied before?

It seems that Deligne's Weil II paper should be relevant here, but alas I don't understand it well enough.

But what I would really like to know is the following.

2. Assume that $\rho$ is irreducible and not the trivial representation. Then does $L(s,\rho)$ admit a holomorphic continuation to the line $\text{re}(s) = n$ without zeros?

This latter fact is proved in the classical case where $n = 1$ by using Brauer's induction theorem to reduce to the case where $G$ is abelian. Here one essentially obtains a dirichlet L-function, where the result is well-known (being part of the proof of Dirichlet's theorem on primes in arithmetic progressions). I imagine a similar reduction to the abelian case can be made in my setting, but of course the point is that I don't know how to deal with the abelian case itself.

Answer 1

According to Serre in "Zeta and L-functions" (MR0194396), these L-functions have been defined by Artin himself. Moreover, it seems that in the formula $$ L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1} $$ one should define $\rho(\mathrm{Frob}_x)$ as the average of the $\rho(g)$ for $g \in G$ mapping to $\mathrm{Frob}_x$.

Serre further says that $L(s,\rho)$ can be continued to a meromorphic function in the half-plane $\mathrm{Re}(s)>n-\frac12$, and that the singularities « can be determined, or rather reduced to the classical case $n=1$ » (there is an interesting lemma in the case $Y \to Y'$ is a fibration of curves, enabling to do induction on the dimension of $Y$). I don't know whether this answers your Question 2.

Artin's paper is Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 292--306 (MR3069563).