The angle of large values of L-functions

We prove three results on the argument of large central values of L  -functions. The first establishes that there exists a sequence of quadratic Dirichlet characters χdχd and Dirichlet polynomials T(χd)T(χd) truncating L(12,χd) at a length a power of d  , such that the truncated sum is large and negative. On the generalized Riemann Hypothesis this distinguishes the central point 12 from fixed σ>12. A result of Kalpokas, Korolev and Steuding establishes large values of the Riemann zeta function among {ζ(12+it):t∈[T,2T]} with prescribed argument modulo π, with a weaker result modulo 2π. Our second result removes the condition modulo π. Our third result proves an analogue in the family of central values of Dirichlet L-functions to fixed prime conductor.