Geodesic
Analytic properties of Euler products for congruence-subgroups of $\operatorname{Sp}_2(\mathbf Z)$
Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 335-363 
S. A. Evdokimov. Analytic properties of Euler products for congruence-subgroups of $\operatorname{Sp}_2(\mathbf Z)$. Sbornik. Mathematics, Tome 38 (1981) no. 3, pp. 335-363. http://geodesic.mathdoc.fr/item/SM_1981_38_3_a1/
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     author = {S. A. Evdokimov},
     title = {Analytic properties of {Euler} products for congruence-subgroups of $\operatorname{Sp}_2(\mathbf Z)$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we prove meromorphic continuation to the entire complex plane and derive a functional equation for the zeta-function $Z_F(s)$ corresponding to a Siegel modular form $F$ which is automorphic for the principal congruence-subgroup of level $q$ in the integral symplectic group $\operatorname{Sp}_2(\mathbf Z)$ of genus $2$ and is an eigenfunction for all of the Hecke operators $T_k(m)$ with index prime to $q$. Bibliography: 9 titles.

[1] S. A. Evdokimov, “Eilerovy proizvedeniya dlya kongruents-podgrupp zigelevoi gruppy roda $2$”, Matem. sb., 99(141) (1976), 483–514 | MR

[2] A. N. Andrianov, “Eilerovy proizvedeniya, otvechayuschie modulyarnym formam Zigelya roda $2$”, Uspekhi matem. nauk, XXIX:3(177) (1974), 43–110 | MR

[3] Seminaire H. Cartan, 10-e annee, 1957–1958, Fonctions Automorphes, Paris, 1958

[4] S. Raghavan, “Modular forms of degree n and representations by quadratic forms”, Ann. Math., 70:3 (1959), 446–477 | DOI | MR

[5] H. Maass, “Die Primzahlen in der Theorie der Siegelsehen Modulfunktionen”, Math. Ann., 124 (1951), 87–122 | DOI | MR | Zbl

[6] E. Hecke, “Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II”, Math. Ann., 114 (1937), 1–28 ; 316–351 | DOI | MR | Zbl | MR | Zbl

[7] G. Shimura, Vvedenie v arifmeticheskuyu teoriyu avtomorfnykh funktsii, izd-vo “Mir”, Moskva, 1973 | MR

[8] J. Kubota, “Über discontinuerliche Gruppen Picardschen Typus und zugehorige Eisensteinsche Reihen”, Nagoya Math. J., 32 (1968), 259–271 | MR | Zbl

[9] S. A. Evdokimov, “O ratsionalnosti proizvodyaschikh ryadov dlya koeffitsientov Fure zigelevykh modulyarnykh form roda $n$”, Analiticheskaya teoriya chisel i teoriya funktsii, Zapiski nauchn. seminarov LOMI, 76, 1978, 65–71 | MR | Zbl