Standard L-function

In mathematics, the term Standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands.
Here standard refers to the finite dimensional representation r being the standard representation of the L-group as a matrix group.

Relations to other L-functions

Standard L-functions are thought to be the most general type of L-function. Conjecturally they include all examples of L-functions, and in particular are expected to coincide with the Selberg class

Selberg class

In mathematics, the Selberg class S is an axiomatic definition of a class of L-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called L-functions or zeta functions...

. Furthermore, all L-functions over arbitrary number fields are widely thought to be instances of standard L-functions for the general linear group GL(n) over the rational numbers Q. This makes them a useful testing ground for statements about L-functions, since it sometimes affords structure from the theory of automorphic forms.

Analytic properties

Those L-functions were proven to always be entire by Godement and Jacquet, with the sole exception of Riemann ζ-function, which arises for n=1. Another proof was later given by Freydoon Shahidi using the Langlands-Shahidi method (see for a useful broader discussion).