Artin's holomorphy conjecture

Let $L/K$ be a Galois extension with Galois group

$$G=\operatorname{Gal}(L/K).$$

For every irreducible character $\chi\colon G\to\mathbb{C}$, let $L(s,\chi)$ denote the associated Artin $L$-function. Artin's holomorphy conjecture. The function $L(s,\chi)$ admits a meromorphic continuation to the entire complex plane, is analytic on $\mathbb{C}$ when $\chi\neq 1$, and is analytic on $\mathbb{C}\setminus\{1\}$ when $\chi=1$. This conjecture asserts the expected holomorphy of nontrivial Artin $L$-functions, with the trivial character allowed its pole at $s=1$; the source gives no resolution status beyond stating the conjecture.