## Siegel zeros and cusp forms.(English)Zbl 0847.11043

The question of whether the $$L$$-function attached to a Dirichlet character has real zeros near 1 (“Siegel zeros”) or not is of great importance in number theory. The study of this classical problem has recently been extended to automorphic $$L$$-functions. It is conjectured that the standard $$L$$-functions of cusp forms on $$\text{GL}(n)$$, $$n\geq 2$$, have no Siegel zeros. In the paper, this conjecture is proved for $$n= 2$$, and for $$n= 3$$ with some restriction. In general it is proved that Siegel zeros are rare (in a certain, well defined, sense) and that they do not exist at all if Langlands’ principle of functoriality is valid for $$\text{GL}(n)\times \text{GL}(n')\to \text{GL}(nn')$$. The proof for $$n= 2,3$$ uses several results on the analytic properties of automorphic $$L$$-functions, namely of the symmetric cube $$L$$-functions of $$\text{GL}(2)$$ and the symmetric square $$L$$-functions of $$\text{GL}(3)$$.

### MSC:

 11M41 Other Dirichlet series and zeta functions 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations
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