Siegel zeros and cusp forms. (English) Zbl 0847.11043

Hoffstein, Jeffrey; Ramakrishnan, Dinakar

Int. Math. Res. Not. 1995, No. 6, 279-308 (1995).

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 \(\text{GL}(3)\).

Reviewer: J.G.M.Mars (Utrecht)

MSC:

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

Keywords:

automorphic \(L\)-functions; \(L\)-functions of cusp forms; Siegel zeros; symmetric cube \(L\)-functions; symmetric square \(L\)-functions