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 representa ...
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 give ris ...
. 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''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are co ...
. 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 f ...
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 invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
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 of ...
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–S ...
using the
Langlands–Shahidi method In mathematics, the Langlands–Shahidi method provides the means to define automorphic L-functions in many cases that arise with connected reductive groups over a number field. This includes Rankin–Selberg products for cuspidal automorphic rep ...
. 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
* ...
References
{{reflist
Zeta and L-functions