A note on the zeros of generalized Hurwitz zeta functions

Given a function $f(n)$ periodic of period $q\geq 1$ and an irrational number $0<\alpha\leq 1$, Chatterjee and Gun proved that the series $F(s,f,\alpha)=\sum_{n=0}^{\infty}\frac{f(n)}{(n+\alpha)^s}$ has infinitely many zeros for $\sigma>1$ when $\alpha$ is transcendental and $F(s,f,\alpha)$ has a pole at $s=1$, or when $\alpha$ is algebraic irrational and $c=\frac{\max{f(n)}}{\min{f(n)}}<1.15$. In this note, we prove that the result holds in full generality.