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 "L""G" 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 formula_1-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).
The L-function formula_2 should be a product over the places formula_3 of formula_4 of local formula_1 functions.
formula_6
Here the automorphic representation formula_7 is a tensor product of the representations formula_8 of local groups.
The L-function is expected to have an analytic continuation as a meromorphic function of all complex formula_9, and satisfy a functional equation
formula_10
where the factor formula_11 is a product of "local constants"
formula_12
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.