L-Functions with n-th-Order Twists

Let K be a number field containing the n-th roots of unity for some n >= 3. 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) with chi and n-th-order Hecke character: an estimate of the second moment on the critical line, a nonvanishing result at the central point, and a zero-density theorem.