There are at least two statements which go by the name of Artin's conjecture.

If r is any complex finite-dimensional representation of the absolute Galois group of a number field, then Artin showed how to associate an L-series with it. These L-series directly generalize zeta functions and Dirichlet L-series, and as a result of work by Richard Brauer, is known to extend to a meromorphic function on the complex plane. Artin's conjecture predicts that it is in fact holomorphic, i.e., has no poles, with the possible exception of a pole at s = 1 (Artin 1923/1924). Compare with the generalized Riemann hypothesis, which deals with the locations of the zeros of certain L-series.

The second conjecture states that every integer not equal to -1 or a square number is a primitive root modulo p for infinitely many p and proposes a density for the set of such p which are always rational multiples of a constant known as Artin's constant. There is an analogous theorem for functions instead of numbers which has been proved by Billharz (Shanks 1993, p. 147).

 

Artin's Constant, Artin L-Function, Generalized Riemann Hypothesis