For $f$ a cuspidal modular form for the group $\Gamma_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive character $e^{ 2\pi i n \frac{ p}{ q}}$ with $\frac{ p}{ q}\in \mathbb{ Q}$. We prove that for certain $f$ and $\frac{ p}{ q}\in \mathbb{ Q}$, the additively twisted $L$-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.
Categories: math.NT