Let $F$ and $E$ be number fields, $G_F$ be the absolute Galois group of $F$, and $S$ be a finite set of primes of $F$. For $\lambda$ a prime of $E$ we denote by $\ell$ its residual characteristic. We also denote the set of primes of $F$ over $\ell$ by $S_{\ell}$.
A compatible system of continuous semisimple $\lambda$-adic representations $\rho=\{\rho_{\lambda}:G_F \to \mathrm{GL}_n(\overline{E_{\lambda}})\}_{\lambda}$ satisfies (by definition) the following properties:
(1) If $v \not\in S \cup S_{\ell}$, then $\rho_{\lambda}$ is rational and unramified at $v$.
(2) If $v \not\in S \cup S_{\ell} \cup S_{\ell'}$, then the characteristic polynomials of the image of $\mathrm{Frob}_v$ under $\rho_{\lambda}$ and $\rho_{\lambda'}$ are the same.
In p.I.16 of Serre's book, 'Abelian $\ell$-adic representations and elliptic curves', it is defined the $L$-function attached to this compatible system by
$$ L(s,\rho):=\prod_{v \not\in S} \frac{1}{\det(1-\rho(\mathrm{Frob}_v)N(v)^{-s})}. $$
Here, the denominator can be taken independently for infinitely many $\lambda$ (precisely, $\lambda$ with $\ell \neq p_v$).
I would like to know whether there are some results related to the analytic properties of this $L$-function, such as the holomorphy or existence of poles(of analytic continuation, if exists), absolute convergence of the formal Dirichlet series on the right half plane, convexity bounds, and similar properties concerning $L(s,\rho \times \rho)$.
If we consider the Artin representation instead of a compatible system of $\lambda$-adic representations, then Artin conjecture states that the associated $L$-function is entire. However, the Artin representation is a complex representation of $G(M/F)$ of finite extension $M$ of $F$. I wonder if is there a similar conjecture or theorem about the $\lambda$-adic representations.
