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