Dirichlet series with periodic coefficients, Riemann's functional equation and real zeros of Dirichlet $L$-functions

In this paper, we give Dirichlet series with periodic coefficients that have Riemann’s functional equation and real zeros of Dirichlet $L$-functions. The details are as follows. Let $L(s,\chi )$ be the Dirichlet $L$-function and $G(\chi )$ be the Gauss sum associate with a primitive Dirichlet character $\chi$ ($\mathrm{m}\mathrm{o}\mathrm{d}q$). Put $f (s,\chi) := q^s L(s,\chi) + i^{-\kappa (\chi)} G(\chi) L(s,\overline{\chi})$,where$\overline{\chi}$ is the complex conjugate of $\chi$ and $\kappa (\chi ):=(1-\chi (-1))\mathrm{/}2$. Then we prove that $f(s,\chi )$ satisfies Riemann’s functional equation appearing in Hamburger’s theorem if $\chi$ is even. In addition, we show that $f(\sigma ,\chi )\mathrm{\ne }0$ all $\sigma \ge 1$. Moreover, we prove that $f(\sigma ,\chi )\mathrm{\ne }0$ for all $1/2 \le \sigma < 1$ifandonlyif$L(\sigma,\chi) \ne 0$forall$1/2 \le \sigma < 1$\mathrm{.}When$\chi$isreal,allzerosof$f(s,\chi)$with$\Re (s) >0$ are on the line $\sigma =1\mathrm{/}2$ if and only if GRH for $L(s,\chi )$ is true. However, $f (s,\chi)$hasinfinitelymanyzerosoffthecriticalline$\sigma =1/2$if$\chi$ is non-real.