Taille des valeurs de fonctions $L$ de carrés symétriques au bord de la bande critique

Abstract

For each weight $k$ and level $N$ square free and without small prime factors, we prove the existence of primitive forms $f_{+}$ and $f_{-}$ of weight $k$ and level $N$ such that

and

The result comes from a delicate study of the moments of $L(1,{\mathrm{sym}}^{2}f)$. This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.