In mathematics, the term standard L-function refers to a particular type of
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 represent ...
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
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...
. Conjecturally, they include all examples of L-functions, and in particular are expected to coincide with the
Selberg class
In mathematics, the Selberg class is an axiomatic definition of a class of L-function, ''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 ...
. Furthermore, all L-functions over arbitrary
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a ...
s are widely thought to be instances of standard L-functions for the
general linear group
In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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 form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...
s.
Analytic properties
These L-functions were proven to always be entire by
Roger Godement
Roger Godement (; 1 October 1921 – 21 July 2016) was a French mathematician, known for his work in functional analysis as well as his expository books.
Biography
Godement started as a student at the École normale supérieure in 1940, where he ...
and
Hervé Jacquet
Hervé Jacquet is a French American mathematician, working in automorphic forms. He is considered one of the founders of the theory of automorphic representations and their associated L-functions, and his results play a central role in modern num ...
, with the sole exception of
Riemann ζ-function, which arises for ''n'' = 1. Another proof was later given by
Freydoon Shahidi
Freydoon Shahidi (born June 19, 1947) is an Iranian American mathematician who is a Distinguished Professor of Mathematics at Purdue University in the U.S. He is known for a method of automorphic L-functions which is now known as the Langlands– ...
using the
Langlands–Shahidi method. For a broader discussion, see .
[.]
See also
*
Zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* A ...
References
{{reflist
Zeta and L-functions