Truncated Sharing of Subsets and Uniqueness of $L$-Functions in the Extended Selberg Class

Let $P(z)$ be a polynomial of degree $q$ without multiple zeros, let $S$ be the zero set of $P(z)$, and let $k$ be the number of distinct roots of the derivative of $P$. Assume that $P(z)$ is a strong uniqueness polynomial for $L$-functions in the Selberg class. We prove that two $L$-functions $L_1$ and $L_2$ in the Selberg class sharing $S$ with multiplicity $\leq m$ (i. e. $E_{L_1,m)}(S)=E_{L_2,m)}(S))$ necessarily coincide if one of the following conditions holds: (i) $m=1$ and $q\geq 2k+5$; (ii) $2\leq m<\infty$ and $q\geq 2k+3$.