  This paper exposes the underlying mechanism for obtaining second integral
moments of $GL_2$ automorphic $L$--functions over an arbitrary number field.
Here, moments for $GL_2$ are presented in a form enabling application of the
structure of adele groups and their representation theory. To the best of our
knowledge, this is the first formulation of integral moments in
adele-group-theoretic terms, distinguishing global and local issues, and
allowing uniform application to number fields. When specialized to the field of
rational numbers $\Bbb{Q}$, we recover the classical results.
