  Let K be a number field containing the n-th roots of unity for some n > 2. We
prove a uniform subconvexity result for a family of double Dirichlet series
built out of central values of Hecke L-functions of n-th order characters of K.
The main new ingredient, possibly of independent interest, is a large sieve for
n-th order characters. As further applications of this tool, we derive several
results concerning L(s,\chi) for n-th order Hecke characters: an estimate of
the second moment on the critical line, a non-vanishing result at the central
point, and a zero-density theorem.
