On Artin's L-Series with General Group Characters

Abstract

where on the right side x is to be interpreted as a character of 5/9L IV. If K is an abelian field over F and if x is an irreducible character, then L(s, x, K/F) coincides with one of the ordinary L-series of the extension field K of F. The proof of this fact rests on the law of reciprocity. The results of Hecke2 show in this abelian case that L(s, X, K/F) is a meromorphic function which satisfies a certain functional equation. Further, Artin proved the group theoretical theorem that every character x is a linear combination E cep, where the c, are rational numbers and where the sp, are characters of @ induced by characters of cyclic subgroups. In connection with I, II, and IV, this yielded at once the result that in the general case L(s, x, K/F) can be continued analytically over the whole complex plane; indeed, a suitable power L(s, x, K/F)m with an integral rational m is a meromorphic function. Moreover, L(s, X, K/F) again satisfies a functional equation of the well known type. However, since m may be larger than 1, this does not show that L(s, x, K/F) itself is a single-valued function. Artin conjectured that this is, in fact, the case and that L(s, x, K/F) is a product of abelian L-series