Standard L-function
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Standard L-function
In mathematics, the term Standard L-function refers to a particular type of automorphic L-function 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. 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
Those L-functions were proven to always be entire by Godement and Jacquet, with the sole exception of Riemann ζ-function, which arises for n=1.
References
Armand Borel, "Automorphic L-functions", Proc. Symp. Pure Math 33, American Mathematical Society, 1979, A. Borel and W. Casselman, editors.
Stephen Gelbart and Freydoon Shahidi, "Analytic Properties of Automorphic L-functions", Academic Press, New York, 1988.
Roger Godement and Herve Jacquet, "Zeta functions of simple algebras", Springer Lecture Notes in Mathematics, volume 260, 1972.