Bounds for the logarithmic derivative in the Selberg Class

Question

Let $F \in \mathcal{S}$, where $\mathcal{S}$ is the set of $L-$functions in the Selberg Class. Are there established upper and lower bounds for $$\left|\frac{F^{'}(s)}{F(s)}\right|,$$ where $s = \sigma + it$?

Any help would be so much appreciated.

Answer 1

For simplicity, assume that $L(s)$ is entire. Writing the completed $L$-function for $L(s)$ as $$ \Lambda(s) = q^{s/2} L(s)\prod_{j=1}^d \Gamma(\lambda_j s + \mu_j), $$ let's define the analytic conductor of $L(s)$ to be $$ C = q \prod_{j=1}^d(|\mu_j|+1)^{\lambda_j}. $$ Using Jensen's formula, we can prove that if $$ \max_{\substack{k\geq 0, \\ k\in\mathbb{Z}}}\Big|s+\frac{k+\mu_j}{\lambda_j}\Big|\geq \frac{1}{2} $$ and $\rho$ ranges over the zeros of $\Lambda(s)$, then $$ \frac{L'}{L}(s) = \sum_{|\rho-s|\leq 1}\frac{1}{s-\rho}+O(\log(C(|s|+3))). $$ Moreover, we know the number of zeros $\rho$ (counting multiplicity) such that $|\rho-s|\leq 1$ is $O(\log(C(|s|+3)))$. Without more information on where the zeros are (e.g., a zero-free region) or how close the nearest zero is to $s$, this is essentially the most that one can say.