  In this paper we provide an explicit construction of a $distinctive$ multiple
Dirichlet series associated to products of quadratic Dirichlet L-series, which
we believe should be tightly connected to a generalized metaplectic Whittaker
function on the double cover of a Kac-Moody group. To do so, we first impose a
set of axioms, independent of any group of functional equations, which the
aforementioned object should satisfy. As a consequence, we deduce that the
coefficients of the $p$-parts of the multiple Dirichlet series satisfy certain
recurrence relations. These relations lead to a family of identities, which
turns out to be $encoded$ in the combinatorial structure of certain moduli
spaces of admissible double covers. Finally, via this crucial connection, we
apply Deligne's theory of weights to express inductively the coefficients of
the $p$-parts in terms of the eigenvalues of Frobenius acting on the
$\ell$-adic \'etale cohomology of local systems on the moduli
$\mathscr{H}_{g}[2]$ of hyperelliptic curves of genus $g$ with level 2
structure.
