Automorphic L-function

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title: "Automorphic L-function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["automorphic-forms", "zeta-and-l-functions", "langlands-program"] description: "Mathematical concept" topic_path: "general/automorphic-forms" source: "https://en.wikipedia.org/wiki/Automorphic_L-function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Mathematical concept ::

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 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.

References

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