Geodesic
Determining $L$-series from their functional equations
Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 537-553 
M. I. Gurevich. Determining $L$-series from their functional equations. Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 537-553. http://geodesic.mathdoc.fr/item/SM_1971_14_4_a4/
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     author = {M. I. Gurevich},
     title = {Determining $L$-series from their functional equations},
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     year = {1971},
     volume = {14},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_4_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we examine a necessary and sufficient condition for a character of the idèles of a number field to belong to the Hecke characters in terms of $L$-series. In addition, we describe all characters of the idèles of $Q$ whose $L$-series have a functional equation and satisfy certain analytic inequalities. Bibliography: 9 titles.

[1] S. Leng, Algebraicheskie chisla, Mir, Moskva, 1966 | MR

[2] I. R. Shafarevich, Dzeta-funktsiya, MGU, Moskva, 1969 | MR

[3] I. M. Gelfand, M. I. Graev, I. I. Pyatetskii-Shapiro, Teoriya predstavlenii i avtomorfnye funktsii, Nauka, Moskva, 1966 | MR

[4] A. Veil, Integrirovanie v topologicheskikh gruppakh i ego primeneniya, IL, Moskva, 1950

[5] G. Veil, Algebraicheskaya teoriya chisel, IL, Moskva, 1947

[6] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Nauka, Moskva, 1968 | MR | Zbl

[7] E. K. Titchmarsh, Vvedenie v teoriyu integralov Fure, Gostekhizdat, Moskva–Leningrad, 1947

[8] E. K. Titchmarsh, Teoriya dzeta-funktsii Rimana, IL, Moskva, 1953

[9] A. Weil, “Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen”, Math. Ann., 158 (1967), 149–156 | DOI | MR