standard L-function
{{Short description|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 = {{mvar|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 {{mvar|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 {{mvar|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 {{harvtxt|Gelbart|Shahidi|1988}}.{{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 {{mvar|L}}-functions
| volume = 6
| year = 1988}}.
See also
References
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