  We study a certain two-parameter family of non-standard graded complete
intersections $A(m,n)$. In case $n=2$, we show that $A(m,2)$ has the strong
Lefschetz property and the complex Hodge-Riemann property if and only if $m$ is
even. This supports a strengthening of a conjecture of Almkvist on the
unimodality of the Hilbert function of $A(m,n)$.
