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The critical order of vanishing of automorphic \(L\)-functions with large level. (English) Zbl 0838.11035
Let \(f\) be a cuspidal newform of weight 2 for the group \(\Gamma_0 (N)\). Then the twisted \(L\)-functions \(L_f (s, \chi)\) are of interest for many reasons, in particular one studies the order of vanishing at the center \(s=1\) of the critical strip of this \(L\)-function. The two main results of this article are: a) If \(\chi\) is a fixed Dirichlet character of conductor \(q\), then there is a constant \(C_q\) and an absolute constant \(C\) such that for \(N\) prime and \(> C_q\) there are at least \(CN \log^{-2} N\) newforms \(f\) for \(\Gamma_0 (N)\) such that \(L_f (1, \chi) \neq 0\). b) If \(\chi_1, \chi_2\) are distinct Dirichlet characters of conductors \(q_1, q_2\) then there are constants \(C_1\), \(C_2\) depending on \(q_1,q_2\) such that for \(N > C_1\) prime there are at least \(C_2 N \log^{-10} N\) newforms \(f\) for \(\Gamma_0 (N)\) for which \(L_f (s, \chi_1) L_f (s, \chi_2)\) has at most a simple zero at \(s = 1\). The conjectured truth in both cases is a positive proportion of \(N\).

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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References:
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