We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime.
Decoupling for perturbed cones and mean square of $|ζ(\frac 12+it)|$
The fourth moment of Dirichlet L-functions
On the mean square of the Riemann zeta-function in short intervals
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