The Zero Density Conjecture for quadratic Dirichlet L-functions over function fields

Let $\varphi(x)$ be a Schwartz class function on $\mathbb{R}$ with Fourier transform having compact support. Define the density

$$W(\mathrm{Sp})(x)dx=\left(1-\frac{\sin(2\pi x)}{2\pi x}\right)dx,$$

and let $\rho=\frac12+i\gamma$, with $\gamma\in\mathbb{C}$, represent the non-trivial zeros of $L(s,\chi_{8d})$. Let $\Lambda(s,\chi_{8d})$ represent the associated completed $L$-function. Zero Density Conjecture. One has

$$\lim_{D\to\infty}\frac1{|s(D)|}\sum_{d\in s(D)}\sum_{\Lambda(\rho,\chi_{8d})=0} \varphi\left(\frac{\gamma\log D}{2\pi}\right)=\int_{-\infty}^{\infty}\varphi(x)W(\mathrm{Sp})(x)dx.$$

This is the density statement for the low-lying zeros in the symplectic family of quadratic Dirichlet $L$-functions. The surrounding discussion presents it as the density conjecture needed, together with the Riemann Hypothesis, to obtain conditional central-limit results; its resolution status is not specified in the supplied text.