Chowla–Chowla nonvanishing conjecture for periodic functions

Let $p$ be a prime and let $f$ be a rational-valued periodic function with period $p$. Define its associated $L$-function, for $[?]s$ with $[?]\operatorname{Re}(s)>1$, by

$$L(s,f)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}.$$

Chowla–Chowla conjecture. $L(2,f)\ne0$, except when

$$f(1)=f(2)=\cdots=f(p-1)=\frac{f(p)}{1-p^2}.$$

This conjecture concerns nonvanishing of special values of $L$-functions attached to arbitrary periodic arithmetic functions and was formulated by P. Chowla and S. Chowla. Its status is not specified in the source.