  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.
