Extreme values of LL‐functions of newforms

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform f of weight k and level one such that L(1/2, f ) ~≥~ exp( (1 + o(1)) √2log k/loglog k ) for sufficiently large k ≡ 0 4. In this note, we show that for any ε>0 and for all sufficiently large k ≡ 0 4, the number of normalized Hecke eigenforms of weight k and level one for which L(1/2, f ) ~≥~ exp(1.41√ log k /loglog k ) is ε k1-ε. For an odd fundamental discriminant D, let Bₖ(|D|) be the set of all cuspidal normalized Hecke eigenforms of weight k and level dividing $|D|$. When the real primitive Dirichlet character χD satisfies χD(-1)= iᵏ, we investigate the number of f ∈ Bₖ(|D|) for which L(1/2, f ⊗ χD) takes extremal values.