  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.
