  We consider a deformation of the two-dimensional free scalar field theory by
raising the Laplacian to a positive real power. It turns out that the resulting
non-local generalized free action is invariant under two commuting actions of
the global conformal symmetry algebra, although it is no longer invariant under
the full Witt algebra. Furthermore, there is an adelic version of this family
of conformal field theories, parameterized by the choice of a number field,
together with a Hecke character. Tate's thesis gives the Green's functions of
these theories, and ensures that these Green's functions satisfy an adelic
product formula. In particular, the local $L$-factors contribute to the
prefactors of these Green's functions. Quadratic reciprocity turns out to be a
consequence of an adelic version of a holomorphic factorization property of
this family of theories on a quadratic extension of $\mathbb{Q}$. We explain
that at the Archimedean place, the desired holomorphic factorization follows
from the global conformal symmetry.
