Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If $R$ is standard graded, we prove that, under the additional hypothesis that $R$ is Koszul or has an irreducible $h$-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.