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. 
@article{FAA_2000_34_2_a2,
     author = {V. A. Bykovskii},
     title = {Functional {Equations} for {Hecke--Maass} {Series}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {23--32},
     publisher = {mathdoc},
     volume = {34},
     number = {2},
     year = {2000},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2000_34_2_a2/}
}
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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/