On the Mahler measure associated to $X_1(13)$

Abstract

We show that the Mahler measure of a defining equation of the modular curve $X_1(13)$ is equal to the derivative at $s=0$ of the $L$-function of a cusp form of weight 2 and level 13 with integral Fourier coefficients. The proof combines Deninger's method, an explicit version of Beilinson's theorem together with an idea of Merel to express the regulator integral as a linear combination of periods. Finally, we present further examples related to the modular curves of level 16, 18 and 25.