The moments of central values of families of L-functions have recently attracted much attention and, with the work of Keating and Snaith [(2000) Commnun. Math. Phys. 214, 57-89 and 91-110], there are now precise conjectures for their limiting values. We develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of L-functions. As an example we work out the case of the family of all Dirichlet L-functions to a prime modulus.
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