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For $f$ a cuspidal modular form for the group $\Gamma_0(N)$ of integral orhalf-integral weight, $N$ a multiple of $4$ in case the weight ishalf-integral, we study the zeros of the $L$-function attached to $f$ twistedby 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 $\frac{p}{q}\in \mathbb{Q}$, theadditively twisted $L$-function has infinitely many zeros on the critical line.We develop a variant of the Hardy-Littlewood method which uses distributions toprove the result.

Infinitely many zeros of additively twisted $L$-functions on the critical line