  For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the
assertion that for every Dirichlet character $\chi$ and all zeros
$\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular,
${\rm GRH}[\frac12]$ is the Generalized Riemann Hypothesis). In this paper, we
show that the validity of ${\rm GRH}[\frac{9}{10}]$ depends only on certain
distributional properties of the zeros of the Riemann zeta function $\zeta(s)$.
No conditions are imposed on the zeros of nonprincipal Dirichlet $L$-functions.
