Geodesic
A note on Linnik's approach to the Dirichlet $L$-functions
Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 123-132

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Let $\chi \pmod q$, $q>1$, be a primitive Dirichlet character. We first present a detailed account of Linnik's deduction of the functional equation of $L(s,\chi )$ from the functional equation of $\zeta (s)$. Then we show that the opposite deduction can be obtained by a suitable modification of the method, involving finer arithmetic arguments. 
@article{TRSPY_2017_296_a8,
     author = {J. Kaczorowski and A. Perelli},
     title = {A note on {Linnik's} approach to the {Dirichlet} $L$-functions},
     journal = {Informatics and Automation},
     pages = {123--132},
     publisher = {mathdoc},
     volume = {296},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a8/}
}
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J. Kaczorowski; A. Perelli. A note on Linnik's approach to the Dirichlet $L$-functions. Informatics and Automation, Analytic and combinatorial number theory, Tome 296 (2017), pp. 123-132. http://geodesic.mathdoc.fr/item/TRSPY_2017_296_a8/