Zeros of Dirichlet series with periodic coefficients

Let $a=(a_n{)}_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic continuation of ${\sum }_{n\ge 1}{a}_n\mathrm{/}{n}^s$, and $N_a({\sigma }_1,{\sigma }_2,T)$ the number of zeros of $F_a(s)$, counted with their multiplicities, in the rectangle ${\sigma }_1<\mathrm{\Re }s<{\sigma }_2$, $\mathrm{\mid }\mathrm{\Im }s\mathrm{\mid }\le T$. We extend previous results of Laurin\v{c}ikas, Kaczorowski, Kulas, and Steuding, by showing that if $F_a(s)$ is not of the form $P(s)L_{\chi }(s)$, where $P(s)$ is a Dirichlet polynomial and $L_{\chi }(s)$ a Dirichlet L-function, then there exists an $\eta =\eta (a)>0$ such that for all $1\mathrm{/}2<{\sigma }_1<{\sigma }_2<1+\eta$, we have $c_{1}T\le N_a({\sigma }_1,{\sigma }_2,T)\le c_{2}T$ for sufficiently large $T$, and suitable positive constants $c_1$ and $c_2$ depending on $a$, ${\sigma }_1$, and ${\sigma }_2$.