Parabolic points and zeta-functions of modular curves

In this paper we obtain explicit formulas for the values at the center of the critical strip of Dirichlet series connected with weight 2 parabolic forms of the group $\Gamma_0(N)$. In particular, these formulas allow us to verify the Birch–Swinnerton-Dyer conjecture on the order of a zero for uniformizable elliptic curves over certain $\Gamma$-extensions. We also give applications to noncommutative reciprocity laws.