Let $(\mathcal{O},\Sigma ,{F}_{\infty})$ be an arithmetic ring of Krull dimension at most 1, $\mathcal{S}=\mathrm{Spec}\mathcal{O}$ and $(\pi :\mathcal{X}\to \mathcal{S};{\sigma}_{1},...,{\sigma}_{n})$ an $n$-pointed stable curve of genus $g$. Write $\mathcal{U}=\mathcal{X}\setminus {\cup}_{j}{\sigma}_{j}\left(\mathcal{S}\right)$. The invertible sheaf ${\omega}_{\mathcal{X}/\mathcal{S}}({\sigma}_{1}+\cdots +{\sigma}_{n})$ inherits a hermitian structure ${\parallel \xb7\parallel}_{\mathrm{hyp}}$ from the dual of the hyperbolic metric on the Riemann surface ${\mathcal{U}}_{\infty}$. In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ${\omega}_{\mathcal{X}/\mathcal{S}}{({\sigma}_{1}+...+{\sigma}_{n})}_{\mathrm{hyp}}$. The theorem is applied to modular curves $X\left(\Gamma \right)$, $\Gamma ={\Gamma}_{0}\left(p\right)$ or ${\Gamma}_{1}\left(p\right)$, $p\ge 11$ prime, with sections given by the cusps. We show ${Z}^{\text{'}}(Y\left(\Gamma \right),1)\sim {e}^{a}{\pi}^{b}{\Gamma}_{2}{(1/2)}^{c}L(0,{\mathcal{M}}_{\Gamma})$, with $p\equiv 11\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}12$ when $\Gamma ={\Gamma}_{0}\left(p\right)$. Here $Z\left(Y\right(\Gamma ),s)$ is the Selberg zeta...