Iwaniec–Luo–Sarnak Zero Density Conjecture for modular and quadratic-character families

Let $\phi(x)$ be a Schwartz-class function on $\mathbb{R}$ whose Fourier transform has compact support. Define

$$W(SO_{\operatorname{even}})(x)\,dx=\left(1+\frac{\sin 2\pi x}{2\pi x}\right)dx,\qquad W(Sp)(x)\,dx=\left(1-\frac{\sin 2\pi x}{2\pi x}\right)dx.$$

For $f\in H_k$, let $\Lambda(s,f)$ be its completed $L$-function, and for $d\in s(D)$ let $\Lambda(s,\chi_{8d})$ be the completed quadratic-character $L$-function. Write their non-trivial zeros as $\rho=\frac12+i\gamma$, allowing $\gamma$ to be complex if the Riemann Hypothesis fails. Iwaniec–Luo–Sarnak's Zero Density Conjecture.

$$\lim_{\substack{k\to\infty\\ k\equiv0\bmod 4}}\frac{1}{|H_k|}\sum_{f\in H_k}\sum_{\substack{\Lambda(\rho,f)=0\rho=\frac12+i\gamma}}\phi\left(\frac{\gamma\log k}{\pi}\right)=\int_{-\infty}^{\infty}\phi(x)W(SO_{\operatorname{even}})(x)\,dx$$

and

$$\lim_{D\to\infty}\frac{1}{|s(D)|}\sum_{d\in s(D)}\sum_{\substack{\Lambda(\rho,\chi_{8d})=0\rho=\frac12+i\gamma}}\phi\left(\frac{\gamma\log D}{2\pi}\right)=\int_{-\infty}^{\infty}\phi(x)W(Sp)(x)\,dx.$$

These formulas assert the predicted orthogonal and symplectic one-level densities for the two families. The source gives no resolution status; such density results are generally known only under restricted support or additional hypotheses.