Murty–Perelli pair-correlation conjecture for the Selberg class

Let $F\in\mathcal S$, let $d_F$ be its degree, let $N_F(T)$ count its positive zero ordinates, and define

$$\mathcal F_F(x,T)=\sum_{0<\gamma,\gamma'\le T}\frac{4x^{i(\gamma-\gamma')}}{4+(\gamma-\gamma')^2},\qquad \mathcal D_F(x,T)=\frac{\mathcal F_F(x,T)}{N_F(T)}.$$

Murty–Perelli pair-correlation conjecture.

$$\mathcal D_F(T^{\theta d_F},T)\sim\theta\quad(0<\theta\le1),\qquad \mathcal D_F(T^{\theta d_F},T)\sim1\quad(\theta\ge1).$$

This predicts a transition near $x=T^{d_F}$ and is attributed to Murty and Perelli. The conjecture is used as the general-$L$-function analogue of pair correlation and remains open.