User:Stephen Miller Math/Automorphic L-functions
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In mathematics, Automorphic L-functions are a particular type of L-function that are attached (still partially conjecturally) to automorphic forms or representations. They conjecturally cover all known constructions of L-functions.
Definition
Like the Riemann Zeta Function, Automorphic L-functions are defined as Euler products over the primes. The local L-factor for the prime p is always the evaluation of a polynomial at 1/p^s, and the factor at infinity is a product of gamma factors, more precisely shifts of Gamma(s) or Gamma(s/2), times a power of pi. The defintion may be recast in terms of the places of a number field, though this description is always valid.
Robert Langlands defined various automorphic L-functions for general automorphic forms in terms of finite dimensional representations of his L-groups.
Primitive L-functions and cusp forms
Often L-functions factor in terms of smaller L-functions. One which does not is called primitive. It was conjectured by Piatetski-Shapiro and others that the primitive L-functions are precisely the L-functions of cuspidal automorphic representations on GL(n) over the rationals Q.