Let $L(\chi, s)$ be the Dirichlet L-Function of a primitive character $\chi$. I believe, if I’m not mistaken, the convergence of the Euler product of $L(\chi, s)$ in the critical strip is known to be related to GRH. In the event that GRH is false for some character $\chi$, then the Euler product diverges for some zero $\rho = \sigma + it$, with $\sigma \geq 1/2$

My question is, does the sum over primes $p$

$\omega = \sum_{p} \frac{\chi(p)}{p^{\rho}}$

diverge in the sense that the real part of $\omega$ blows up and goes to $-\infty$, or is the real part of the partial sum oscillatory and bounded, or is it oscillatory and unbounded? My hunch has been that it blows up and goes to $-\infty$ with only finitely many sign changes.

And in the event that the zeros of $L(\chi, s)$ cannot get too close to the line $\text{Re}(s) = 1$, that is, there is a vertical strip within the critical strip that acts as a buffer zone between said line and the non-trivial zeros, would the Euler product converge in this buffer zone?