Bounds for moments of Dirichlet L-functions to a fixed modulus

We study the 2k-th moment of central values of the family of Dirichlet L-functions to a fixed prime modulus. We establish sharp lower bounds for all real \(k \ge 0\) $k\ge 0$ and sharp upper bounds for k in the range \(0 \le k \le 1\) $0\le k\le 1$ .