# The twisted second moment of modular half integral weight $L$--functions.

@article{Dunn2019TheTS, title={The twisted second moment of modular half integral weight \$L\$--functions.}, author={Alexander Dunn and Alexandru Zaharescu}, journal={arXiv: Number Theory}, year={2019} }

Given a half-integral weight holomorphic Kohnen newform $f$ on $\Gamma_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}} \big | L(1/2,f,\chi) \big |^2. \end{equation*} Our result is unconditional, it does not rely on the Ramanujan-Petersson conjecture for the form $f$. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution…

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