Journal of the
Korean Mathematical Society JKMS

Let $f$ be a self-dual primitive Maass or modular forms for level $4$. For such a form $f$, we define \begin{align*} N_f^s(T)\!:=\!|\{\rho \in \mathbb{C} : |\Im(\rho)| \leq T, \text{ $\rho$ is a non-trivial simple zero of $L_f(s)$} \}|. \end{align*} We establish an omega result for $N_f^s(T)$, which is $N_f^s(T)=\Omega \big( T^{\frac{1}{6}-\epsilon} \big)$ for any $\epsilon>0$. For this purpose, we need to establish the Weyl-type subconvexity for $L$-functions attached to primitive Maass forms by following a recent work of Aggarwal, Holowinsky, Lin, and Qi.

Keywords: Simple zero, Maass forms, Weyl-type subconvexity

MSC numbers: Primary 11M26, 11L07

Supported by: Peter J. Cho is supported by the NRF grant funded by the Korea government(MSIT) (No. 2019R1F1A1062599) and the Basic Science Research Program(2020R1A4A1016649).