Geodesic
Functional Equations for Hecke--Maass Series
Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 2, pp. 23-32.

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The Dirichlet (Hecke–Maass) series associated with the eigenfunctions $f$ and $g$ of the invariant differential operator $\Delta_k=-y^2(\partial^2\!/\partial x^2+\partial^2\!/\partial y^2)+ iky\,\partial/\partial x$ of weight $k$ are investigated. It is proved that any relation of the form $(f|_kM)=g$ for the $k$-action of the group $SL_2(\mathbb{R})$ is equivalent to a pair of functional equations relating the Hecke–Maass series for $f$ and $g$ and involving only traditional gamma factors. 
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V. A. Bykovskii. Functional Equations for Hecke--Maass Series. Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 2, pp. 23-32. http://geodesic.mathdoc.fr/item/FAA_2000_34_2_a2/

[1] Kuznetsov N. V., Gipoteza Peterssona dlya form vesa nul i gipoteza Linnika, Preprint, Khabarovsk, 1977 | Zbl

[2] Goldfeld D., “Analytic and arithmetic theory of Poincare series”, Journees Arithmetiques de Luminy, Colloq. Internat. CNRS (Centre Univ. Luminy, Luminy, 1978), Asterisque, 61, Soc. Math. France, Paris, 1979, 95–107 | MR

[3] Hecke E., “Über die Bestimmung Dirichletscher Reihen durch ihre Functionalgleichung”, Math. Ann., 112 (1936), 644–669 ; Mathematische Werke, Gottingen, 1959, 591–626 | DOI | MR | MR

[4] Ogg A., Modular forms and Dirichlet series, Benjamin, New York–Amsterdam, 1969 | MR | Zbl

[5] Maass H., Lectures on modular functions of one complex variable, Tata Institute Lecture Notes, 29, Springer-Verlag, 1983 | MR | Zbl

[6] Goldfeld D., Hoffstein J., “Eisenstein series of $1/2$-integral weight and the mean value of real Dirichlet L-series”, Invent. Math., 80 (1985), 185–208 | DOI | MR | Zbl

[7] Bump D., Friedberg S., Hoffstein J., “On some applications of automorphic forms to number theory”, Bull. Amer. Math. Soc., 33:2 (1996), 157—175 | DOI | MR | Zbl

[8] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. I, Nauka, M., 1973 | MR

[9] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. II, Nauka, M., 1966 | MR

[10] Lyuk Yu., Spetsialnye matematicheskie funktsii i ikh approksimatsii, Mir, M., 1980

[11] Sleiter L., Vyrozhdennye gipergeometricheskie funktsii, Bibl. mat. tablits, vyp. 39, M., 1966 | MR