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.
- 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.
- 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.