  Let $q$ be a positive integer. In our recent paper, we proved that the
cardinality of the complement of an integral arrangement, after the modulo $q$
reduction, is a quasi-polynomial of $q$, which we call the characteristic
quasi-polynomial. In this paper, we study general properties of the
characteristic quasi-polynomial as well as discuss two important examples: the
arrangements of reflecting hyperplanes arising from irreducible root systems
and the mid-hyperplane arrangements. In the root system case, we present a
beautiful formula for the generating function of the characteristic
quasi-polynomial which has been essentially obtained by Ch. Athanasiadis and by
A. Blass and B. Sagan. On the other hand, it is hard to find the generating
function of the characteristic quasi-polynomial in the mid-hyperplane
arrangement case. We determine them when the dimension is less than six.
