Universality conjecture for low-lying zeros of automorphic families

Let $\mathfrak{F}$ be a rank $0$ essentially homogeneous family. Write the zeros of each completed $L$-function as $\frac12+i\gamma_j^{(\pi)}$ and normalize them by $\widetilde{\gamma}_j^{(\pi)}=\gamma_j^{(\pi)}\log C(\pi)/(2\pi)$. Let $W_0^{(r)}$, $W_-^{(r)}$, and $W_+^{(r)}$ denote the fluctuation $r$-level densities associated with the four limiting random-matrix symmetry types. Universality conjecture. The low-lying zeros follow the table: non-self-dual families have type $\mathrm{U}(\infty)$ and density $W_0^{(r)}$; orthogonal families have type $\operatorname{Sp}(\infty)$ and density $W_-^{(r)}$; symplectic families with $\varepsilon=1$ have type $\operatorname{SO}_{\mathrm{even}}(\infty)$ and density $W_+^{(r)}$; and symplectic families with $\varepsilon=-1$ have type $\operatorname{SO}_{\mathrm{odd}}(\infty)$ and density $W_-^{(r)}$, in each case for $r\geq 1$. This is the low-lying-zero form of the expected Katz–Sarnak universality law; the source does not state a general proof.