For large prime P and T at least on the order of 1 over sqrt(log P), the summed count of low-lying zeros on the critical line over characters mod P satisfies sum N0(T, chi) much greater than T squared P sqrt(log P).
Selberg, On the zeros of Riemann’s zeta-function,Skr
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Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions
For large prime P and T at least on the order of 1 over sqrt(log P), the summed count of low-lying zeros on the critical line over characters mod P satisfies sum N0(T, chi) much greater than T squared P sqrt(log P).