Artin conjecture

There are other Artin conjectures, but they are not within the scope of this wiki

Statement
The Artin conjecture on Artin L-functions states that the Artin L-function $$L(\rho,s)$$ is analytic (viz, holomorphic) in the whole complex plane, for any nontrivial irreducible Galois representation $$\rho$$.

True for one-dimensional representations
The Artin conjecture is true for one-dimensional representations of the Galois group. In this case, the Artin L-function reduces to the Hecke L-function.

True for two-dimensional representations
The Artin conjecture has been settled for all two-dimensional representations:


 * The cyclic and dihedral case follow from Hecke's work
 * Langlands did the tetrahedral case
 * Tunnell did the octahedral case, extending work of Langlands

Meromorphicity
The Brauer induction theorem states that every character of a finite group occurs as a rational linear combination of characters induced from cyclic groups. This along with Galois theory, shows that all the Artin L-functions are meromorphic.

Proof in the function fields case
The Artin conjecture has been settled in the case of function fields.