Automorphic L-function

In mathematics, an automorphic L-function is a function L(s,π,r) of a complex variable s, associated to an automorphic representation π of a reductive group G over a global field and a finite-dimensional complex representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by.

and gave surveys of automorphic L-functions.

Properties
Automorphic $$L$$-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function $$L(s, \pi, r)$$ should be a product over the places $$v$$ of $$F$$ of local $$L$$ functions.

$$ L(s, \pi, r) = \prod_v L(s, \pi_v, r_v)$$

Here the automorphic representation $$\pi = \otimes\pi_v$$ is a tensor product of the representations $$\pi_v$$ of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex $$s$$, and satisfy a functional equation

$$ L(s, \pi, r) = \epsilon(s, \pi, r) L(1 - s, \pi, r^\lor)$$

where the factor $$\epsilon(s, \pi, r)$$ is a product of "local constants"

$$\epsilon(s, \pi, r) = \prod_v \epsilon(s, \pi_v, r_v, \psi_v) $$

almost all of which are 1.

General linear groups
constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.

In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.