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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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