Geodesic
On the analytic properties of standard zeta functions of siegel modular forms
Sbornik. Mathematics, Tome 35 (1979) no. 1, pp. 1-17 
A. N. Andrianov; V. L. Kalinin. On the analytic properties of standard zeta functions of siegel modular forms. Sbornik. Mathematics, Tome 35 (1979) no. 1, pp. 1-17. http://geodesic.mathdoc.fr/item/SM_1979_35_1_a0/
@article{SM_1979_35_1_a0,
     author = {A. N. Andrianov and V. L. Kalinin},
     title = {On the analytic properties of standard zeta functions of siegel modular forms},
     journal = {Sbornik. Mathematics},
     pages = {1--17},
     year = {1979},
     volume = {35},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1979_35_1_a0/}
}
TY  - JOUR
AU  - A. N. Andrianov
AU  - V. L. Kalinin
TI  - On the analytic properties of standard zeta functions of siegel modular forms
JO  - Sbornik. Mathematics
PY  - 1979
SP  - 1
EP  - 17
VL  - 35
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1979_35_1_a0/
LA  - en
ID  - SM_1979_35_1_a0
ER  - 
%0 Journal Article
%A A. N. Andrianov
%A V. L. Kalinin
%T On the analytic properties of standard zeta functions of siegel modular forms
%J Sbornik. Mathematics
%D 1979
%P 1-17
%V 35
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1979_35_1_a0/
%G en
%F SM_1979_35_1_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that standard zeta functions (analogs of the zeta functions of Rankin and Shimura) for holomorphic cusp forms with respect to congruence subgroups of the form $$ \Gamma_0^n(q)=\biggl\{\begin{pmatrix}A&B\\C&D\end{pmatrix}\in Sp_n(\mathbf Z);\quad C\equiv0\pmod q\biggr\} $$ of the Siegel modular group $Sp_n(\mathbf Z)$ of arbitrary even degree $n$ have a meromorphic continuation. For the case $q=1$, with some additional restrictions, it is proved that the zeta functions are holomorphic except for a finite number of poles, and a functional equation is obtained. Bibliography: 9 titles.

[1] A. N. Andrianov, “Eilerovy razlozheniya teta-preobrazovanii zigelevykh modulyarnykh form roda $n$”, Matem. cb., 105(147) (1978), 291–342 | MR

[2] A. N. Andrianov, “Simmetricheskie kvadraty dzeta-funktsii zigelevykh modulyarnykh form roda 2”, Trudy Matem. in-ta im. V. A. Steklova, CXLII, 1976, 22–45 | MR | Zbl

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

[4] V. L. Kalinin, “Ryady Eizenshteina na simplekticheskoi gruppe”, Matem. sb., 103(145) (1977), 519–549 | MR | Zbl

[5] R. P. Lenglends, “Eilerovy proizvedeniya”, Matematika, 15:1 (1971), 14–43 | MR | Zbl

[6] Kharish-Chandra, Avtomorfnye formy na poluprostykh gruppakh Li, izd-vo “Mir”, Moskva, 1971 | MR

[7] E. Freitag, “Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modulgruppe zweiten Grades”, Inv. Math., 30 (1975), 181–196 | DOI | MR | Zbl

[8] H. Maass, “Siegel's Modular Forms and Dirichlet Series”, Lecture Notes in Math., 216, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[9] H. L. Resnikoff, “Automorphic forms of singular weight are singular forms”, Math. Ann., 215 (1975), 173–193 | DOI | MR | Zbl