Langlands' strong Artin reciprocity conjecture

Let $L/k$ be a finite Galois extension of number fields with $\operatorname{Gal}(L/k)\simeq G$, and let $\rho$ be an $m$-dimensional complex representation of $G$. Let $L(s,\rho)$ and $L(s,\pi)$ denote the Artin and automorphic $L$-functions, respectively. Strong Artin conjecture. There exists an automorphic representation $\pi(\rho)$ of $\mathrm{GL}_m(\mathbb A_k)$ such that, outside finitely many places $v$,

$$L(s,\rho_v)=L(s,\pi_v).$$

Moreover, if $\rho$ is irreducible, then $\pi(\rho)$ is cuspidal. This is a form of Langlands reciprocity and remains open in general.