Small zeros of Dirichlet $L$-functions of quadratic characters of prime modulus

In this paper, we investigate the distribution of the imaginary parts of zeros near the real axis of Dirichlet $L$-functions associated to the quadratic characters $\chi_{p}(\cdot)=(\cdot |p)$ with $p$ a prime number. Assuming the Generalized Riemann Hypothesis (GRH), we compute the one-level density for the zeros of this family of $L$-functions under the condition that the Fourier transform of the test function is supported on a closed subinterval of $(-1,1)$. We also write down the ratios conjecture for this family of $L$-functions a la Conrey, Farmer and Zirnbauer and derive a conjecture for the one-level density which is consistent with the main theorem of this paper and with the Katz-Sarnak prediction and includes lower order terms. Following the methods of \"Ozl\"uk and Snyder, we prove that GRH implies $L(\frac{1}{2},\chi_p)\neq 0$ for at least $75\%$ of the primes.