Geodesic
Determining $L$-series from their functional equations
Sbornik. Mathematics, Tome 14 (1971) no. 4, pp. 537-553

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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. 
@article{SM_1971_14_4_a4,
     author = {M. I. Gurevich},
     title = {Determining $L$-series from their functional equations},
     journal = {Sbornik. Mathematics},
     pages = {537--553},
     publisher = {mathdoc},
     volume = {14},
     number = {4},
     year = {1971},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_14_4_a4/}
}
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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/