  In this paper, we study the antipode of a finite-dimensional Hopf algebra $H$
with the dual Chevalley property and obtain an annihilation polynomial for its
antipode $S$. The annihilation polynomial is determined by the exponent $N$ of
the coradical and the Loewy length. In particular, the order of $S^2$ divides
$N$ in characteristic $0$. Moreover, we get two characterizations of the
quasi-exponent.
