The Euler-product characterization of GRH for nonprincipal Dirichlet L-functions

Let $\chi$ be a primitive nonprincipal Dirichlet character, and let $L(s,\chi)$ be its Dirichlet $L$-function. For each $s$ with ${\rm Re}(s)>1/2$, consider the ordered partial Euler products over primes $p\le n$.

Euler-product characterization of GRH. If $\chi\ne\1$, then

$$L(s,\chi)=\lim_{n\to\infty}\prod_{p\le n}(1-\chi(p)p^{-s})^{-1},$$

where the product is taken over all primes $p$ satisfying $p\le n$.

The source states that, for $\chi\ne\1$, this is equivalent to the generalized Riemann hypothesis. Since the products are not absolutely convergent, their ordering by increasing primes is essential.