Small circulant complex Hadamard matrices of Butson type

We study the circulant complex Hadamard matrices of order nn whose entries are llth roots of unity. For n=ln=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pqn=p+q,l=pq with p,qp,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,ln,l.