Factor of the Euler Product at the Roots Of Zeta

Question

The $\zeta$ function maybe written as Euler Product: $$ \zeta(s)=\prod_{p} \frac{1}{1-p^{-s}}=\prod_p e_p(s). $$ Now let's substitute $s$ with $\rho_k$, the $k$th root of $\zeta$, and have a look at the individual factors. There are 2 options: 1.$|e_p(\rho_k)|<1$ or 2. $|e_p(\rho_k)|\ge1$ (e.g. $|e_{23}(\rho_1)|\approx 1.2404$, for more values, see here) and obviously the values less $1$ must be infintely many more than the others, otherwise it would not converge to $0$. So we can write it as: $$ \zeta(\rho_k)=\prod_{\color{blue}{||<1}} e_p(\rho_k) \times \prod_{\color{red}{||\geq 1}} e_p(\rho_k), $$

So my questions are:

2. Do these values $e_p(\rho_k)$ have a special meaning or a straight forward interpretation?
3. For a given $\rho_k$, how do these values distribute? Plots for $\rho_1$ and $\rho_2$ (not shown) show a spiral around $1+0i$: ($x$ is real axis, $y$ the imaginary, the line indicates $||=1$)
4. obsolete Are there finitely or infinitely many primes $p$, where $|e_p(\rho_k)|\ge1$? Infinitely.
5. How does that distribution behave, if $\Re(s)\neq \frac{1}{2}$? Some values of $e_p(\varepsilon + \rho_k)$ should move outwards, such that $\prod_{||<1} e_p(\rho_k)$ doesn't become $0$ anylonger, if Riemann's Hypothesis is true. $s=1/8+14.134725i$ is shown in the plot:

Thanks for your help/comments/plots/answers...

Answer 1

In this MO question, I showed that, if $s = \sigma+i t$ with $\sigma \in (0,1)$ and $t$ nonzero, then the partial products of the Euler product oscillate extremely wildly.

Let $\Pi_P$ be $\prod_{p \leq P} e_p(s)$. I show that, for any $M> 1$, there are arbitrarily many $P_1 < P_2$ such that $|\Pi_{P_2}| > M | \Pi_{P_1} |$ and infinitely many $P_1 < P_2$ such that $|\Pi_{P_2}| < M^{-1} | \Pi_{P_1} |$. In terms of your metaphor of driving inwards and driving outwards, the product is driven in both directions for very long trips.

This means that numerical data on these partial products is untrustworthy. I suspect that they do not approach zero even at the zeroes of $\zeta$, although I can't prove that. GH (in the same question) proved that, assuming RH, the partial products don't approach zero when $\sigma>1/2$.

It would probably be good to nail down what these products do on the critical line, but it looks to me like they don't have much to do with the behavior of the $\zeta$ function.