  In this paper, we compute the E-polynomials of the
$PGL(2,\mathbb{C})$-character varieties associated to surfaces of genus $g$
with one puncture, for any holonomy around it, and compare it with its
Langlands dual case, $SL(2,\mathbb{C})$. The study is based on the
stratification of the space of representations and on the analysis of the
behaviour of the E-polynomial under fibrations.
