Spectra of Beurling algebras of abelian groups

We give a complete classification of all sets which are spectra of Beurling algebras of any locally compact abelian group, parametrizing them by compact convex sets in the dual of a certain locally convex vector space generated by the group. Hence, for example, a finitely generated free abelian group may enjoy a more complicated range of such spectra than the does rational numbers.