  Let $N\geq 3$ be a composite, odd, and square-free integer and let $\Gamma$
be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular
curve of genus $g_{\Gamma}$ associated to $\Gamma$. In this article, we study
the Arakelov invariant $e(\Gamma)=\bar{\omega}^2/\varphi(N)$, with
$\bar{\omega}^2$ denoting the self-intersection of the relative dualizing sheaf
for the minimal regular model of $X(N)$, equipped with the Arakelov metric, and
$\varphi(N)$ is the Euler's phi function. Our main result is the asymptotics
$e(\Gamma) = 2g_{\Gamma}\log(N) + o(g_{\Gamma}\log(N))$, as the level $N$ tends
to infinity.
