We call a quadratic character $\chi$ mod $d$ positive definite if its partial sums at all different truncations are nonnegative. It is known that such characters are rare due to an earlier result of Baker, Montegomery and a recent result of Kalmynin. We give a new upper bound, which is conjectured to be close to optimal based on a probabilistic model. The proof is based on establishing a $L$-function analogue of Fydorov--Hiary--Keating conjecture.

This is joint work in progress with R. Angelo and K. Soundararajan.