I did calculations using (simple reasonings with analysis and) the properties of the reference to get two lemmas.
Question. Do you recognize if in my deductions and calculations there were mistakes? That is if some of my lemmas is wrong, then you can tell me where could be some mistake, and if the lemmas are rights provide me hints to do a comparison with my calculations. Many thanks.
Let $N\geq 1$ a fixed integer and we define the complex funtion $$f_N(z)=\prod_{n=1}^N e^{-(\rho_n-\frac{1}{2})(\frac{1}{2}-\rho_n)iz},$$ where we are denoting with $\rho_n$ a sequence of complex numbers with $\rho_n=\sigma_n+it_n$, that is $\Re\rho_n=\sigma_n$ and imaginary part $\Im\rho_n=t_n$, satisfying the conditions $0<\sigma_n<1$ and $0<t_n$, I mean when (for each integer $n$ in this segment) $1\leq n\leq N$.
Lemma 1. $\forall N\geq 1$ the function $f_N(z)$ belongs to the Pólya class.
And this
Lemma 2. If for each $1\leq n\leq N$, I say for all integers $n$ in our segment, the condtion $\Re\rho_n=\frac{1}{2}$ holds, then $$\Re\left(\frac{f'_N(z)}{f_N(z)}\right)=0.$$