  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.
