We consider a multi-parameter family of canonical coordinates and mirror maps originally introduced by Zudilin. This family includes many of the known one-variable mirror maps as special cases, in particular many of modular origin and the celebrated ‘quintic’ example of Candelas, de la Ossa, Green and Parkes. In a previous paper, we proved that all coefficients in the Taylor expansions at 0 of these canonical coordinates (and, hence, of the corresponding mirror maps) are integers. Here we prove that all coefficients in their Taylor expansions at 0 are positive. Furthermore, we provide several results about the behaviour of the canonical coordinates and mirror maps as complex functions. In particular, we address their analytic continuation, points of singularity, and radius of covergence. We present several very precise conjectures on the radius of covergence of the mirror maps and the sign pattern of the coefficients in their Taylor expansions at 0.