  In this paper we are interested in moments of Minkowski question mark
function ?(x). It appears that, to certain extent, the results are analogous to
the results obtained for objects associated with Maass wave forms: period
functions, L-series, distributions, spectral properties. These objects can be
naturally defined for ?(x) as well. Despite the fact that there are various
nice results about the nature of ?(x), these investigations are mainly
motivated from the perspective of metric number theory, Hausdorff dimension,
singularity and generalizations. In this work it is shown that analytic and
spectral properties of various integral transforms of ?(x) do reveal
significant information about the question mark function. We prove asymptotic
and structural results about the moments, calculate certain integrals involving
?(x), define an associated zeta function, generating functions, Fourier series,
and establish intrinsic relations among these objects. At the end of the paper
it is shown that certain object associated with ?(x) establish a bridge between
realms of imaginary and real quadratic irrationals.
