Artin's holomorphy conjecture for nontrivial representations

Let $L/k$ be a finite normal extension of number fields with $\operatorname{Gal}(L/k)\simeq G$, let $\rho$ be a complex representation of $G$, and let $L(s,\rho,L/k)$ be the associated meromorphic Artin $L$-function. Artin's holomorphy conjecture. If $\rho$ does not contain the trivial representation, then

$$L(s,\rho,L/k)$$

is entire. This is a central open conjecture on Artin $L$-functions; it strengthens the known holomorphy results beyond the abelian setting.