  The six-derivative conformal scalar operator was originally found by Hamada
in its critical dimension of spacetime, $d=6$. We generalize this construction
to arbitrary dimensions $d$ by adding new terms cubic in gravitational
curvatures and by changing its coefficients of expansion in various curvature
terms. The consequences of global scale-invariance and of infinitesimal local
conformal transformations are derived for the form of this generalized
operator. The system of linear equations for coefficients is solved giving
explicitly the conformal Hamada operator in any $d$. Some singularities in
construction for dimensions $d=2$ and $d=4$ are noticed. We also prove a
general theorem that a scalar conformal operator with $n$ derivatives in
$d=n-2$ dimensions is impossible to construct. Finally, we compare our explicit
construction with the one that uses conformal covariant derivatives and
conformal curvature tensors. We present new results for operators built with
different orders of conformal covariant derivatives.
