Standard L-function

In mathematics, the term standard L-function refers to a particular type of automorphic L-function described by Robert P. Langlands.[1][2] 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,[3] 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 Gelbart & Shahidi (1988).[4]

See also

• Zeta function

References

1. Langlands, R.P. (1978), L-Functions and Automorphic Representations (ICM report at Helsinki) (PDF).
2. Borel, A. (1979), "Automorphic L-functions", Automorphic forms, representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: Amer. Math. Soc., pp. 27–61, MR 0546608.
3. Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lecture Notes in Mathematics, 260, Berlin-New York: Springer-Verlag, MR 0342495.
4. Gelbart, Stephen; Shahidi, Freydoon (1988), Analytic properties of automorphic L-functions, Perspectives in Mathematics, 6, Boston, MA: Academic Press, Inc., ISBN 0-12-279175-4, MR 0951897.