  Let $a=(a_n)_{n\ge 1}$ be a periodic sequence, $F_a(s)$ the meromorphic
continuation of $\sum_{n\ge 1} a_n/n^s$, and $N_a(\sigma_1, \sigma_2, T)$ the
number of zeros of $F_a(s)$, counted with their multiplicities, in the
rectangle $\sigma_1 < \Re s < \sigma_2$, $|\Im s | \le T$. 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 $1/2 < \sigma_1 < \sigma_2 < 1+\eta$, 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 $\sigma_2$.
