Standard L-function

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

::summary Mathematical concept ::

In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands.{{citation | last = Langlands | first = R.P. | title = L-Functions and Automorphic Representations (ICM report at Helsinki) | url = http://publications.ias.edu/sites/default/files/lfunct-ps.pdf | year = 1978}}.{{citation | last = Borel | first = A. | contribution = Automorphic L-functions | mr = 546608 | pages = 27–61 | publisher = Amer. Math. Soc. | location = Providence, R.I. | series = Proc. Sympos. Pure Math. | volume = XXXIII | title = Automorphic forms, representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2 | year = 1979}}. 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. 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

These L-functions were proven to always be entire by Roger Godement and Hervé Jacquet,{{citation | last1 = Godement | first1 = Roger | author1-link = Roger Godement | last2 = Jacquet | first2 = Hervé | author2-link = Hervé Jacquet | mr = 0342495 | publisher = Springer-Verlag | location = Berlin-New York | series = Lecture Notes in Mathematics | volume = 260 | title = Zeta functions of simple algebras | year = 1972}}. 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. For a broader discussion, see .{{citation | last1 = Gelbart | first1 = Stephen | author1-link = Stephen Gelbart | last2 = Shahidi | first2 = Freydoon | author2-link = Freydoon Shahidi | isbn = 0-12-279175-4 | mr = 951897 | publisher = Academic Press, Inc. | location = Boston, MA | series = Perspectives in Mathematics | title = Analytic properties of automorphic L-functions | volume = 6 | year = 1988}}.

References

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