Some properties of Dirichlet $L$-functions associated with their nontrivial zeros

The connection between the values of the $L$-functions at the point $s=1/2$ and Gauss sums which leads to the sufficient condition for the validity of the equality $L(1/2,\chi)=0$ is studied. The necessary conditions for the validity of the Extended Riemann Hypothesis for the $L$-fuctions are given in terms of the signs of the even-order derivatives of the fuctionn $\xi(s,\chi)$ which is an analogue of the Riemann $\xi$-function $\xi(s)$. All the results are applied to the $L$-functions $L(s,\chi)$ with a character $\chi$ being equal to a Legendre symbol.