  We study analytic and arithmetic properties of the elliptic gamma function $$
\prod_{m,n=0}^\infty\frac{1-x^{-1}q^{m+1}p^{n+1}}{1-xq^mp^n}, \qquad |q|,|p|<1,
$$ in the regime $p=q$; in particular, its connection with the elliptic
dilogarithm and a formula of S. Bloch. We further extend the results to more
general products by linking them to non-holomorphic Eistenstein series and, via
some formulae of D. Zagier, to elliptic polylogarithms.
