A note on Linnik's approach to the Dirichlet $L$-functions

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.