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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 representation r of the Langlands dual group LG of G, generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by Langlands (1967, 1970, 1971).

Borel (1979) and Arthur & Gelbart (1991) gave surveys of automorphic L-functions.
Properties

Automorphic L-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).

The L-function \( {\displaystyle L(s,\pi ,r)} \) should be a product over the places v of F of local L functions.

\( {\displaystyle L(s,\pi ,r)=\Pi L(s,\pi _{v},r_{v})} \)

Here the automorphic representation \( {\displaystyle \pi =\otimes \pi _{v}} \) is a tensor product of the representations \( \pi _{v} \) of local groups.

The L-function is expected to have an analytic continuation as a meromorphic function of all complex s, and satisfy a functional equation

\( {\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })} \)

where the factor \( {\displaystyle \epsilon (s,\pi ,r)} \) is a product of "local constants"

\( {\displaystyle \epsilon (s,\pi ,r)=\Pi \epsilon (s,\pi _{v},r_{v},\psi _{v})} \)

almost all of which are 1.
General linear groups

Godement & Jacquet (1972) constructed the automorphic L-functions for general linear groups with r the standard representation (so-called standard L-functions) and verified analytic continuation and the functional equation, by using a generalization of the method in Tate's thesis. Ubiquitous in the Langlands Program are Rankin-Selberg products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the Langlands–Shahidi method.

In general, the Langlands functoriality conjectures imply that automorphic L-functions of a connected reductive group are equal to products of automorphic L-functions of general linear groups. A proof of Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.

References
Arthur, James; Gelbart, Stephen (1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.), L-functions and arithmetic (Durham, 1989) (PDF), London Math. Soc. Lecture Note Ser., 153, Cambridge University Press, pp. 1–59, doi:10.1017/CBO9780511526053.003, ISBN 978-0-521-38619-7, MR 1110389
Borel, Armand (1979), "Automorphic L-functions", in Borel, Armand; Casselman, W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 27–61, doi:10.1090/pspum/033.2/546608, ISBN 978-0-8218-1437-6, MR 0546608
Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004), Lectures on automorphic L-functions, Fields Institute Monographs, 20, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3516-6, MR 2071722
Gelbart, Stephen; Piatetski-Shapiro, Ilya; Rallis, Stephen (1987), Explicit constructions of automorphic L-functions, Lecture Notes in Mathematics, 1254, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0078125, ISBN 978-3-540-17848-4, MR 0892097
Godement, Roger; Jacquet, Hervé (1972), Zeta functions of simple algebras, Lecture Notes in Mathematics, 260, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0070263, ISBN 978-3-540-05797-0, MR 0342495
Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg Convolutions", Amer. J. Math., 105: 367–464, doi:10.2307/2374264
Langlands, Robert (1967), Letter to Prof. Weil
Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math, 170, Berlin, New York: Springer-Verlag, pp. 18–61, doi:10.1007/BFb0079065, ISBN 978-3-540-05284-5, MR 0302614
Langlands, Robert P. (1971) [1967], Euler products, Yale University Press, ISBN 978-0-300-01395-5, MR 0419366
Shahidi, F. (1981), "On certain "L"-functions", Amer. J. Math., 103: 297–355, doi:10.2307/2374219

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