Remarks on a special value of the Selberg zeta function

Let $\CmZ_{Y_0(N)}$ be the constant term of the logarithmic derivative at $s=1$ of the Selberg zeta function of the modular curve $Y_0(N)$. Jorgenson and Kramer established the bound $\CmZ_{Y_0(N)}=O_\epsilon(N^\epsilon)$, $\epsilon>0$ by relating it to geometric invariants. In this article we give, for $N$ prime, another proof via $L$-functions and exponential sums improving on a previous approach by Abbes-Ullmo and Michel-Ullmo. We further derive a power of $\log N$ bound along the same line.