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Zeros of linear combinations of Hecke \( L\)-functions on the critical line. (English. Russian original) Zbl 1233.11098
Izv. Math. 74, No. 6, 1277-1314 (2010); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 6, 183-222 (2010).
Let \(\psi_j\) be the characters of the ideal class group of an imaginary quadratic field \({\mathbb Q}(\sqrt{-D})\), \(L(s,\psi_j)\) the corresponding Hecke \(L\)-functions, \(a_j\) real numbers. The author proves: If \(T\) is sufficiently large, then the number of odd-order zeros of \(\sum_j a_jL(s,\psi_j)\) in the interval \(\{s=\frac 12+it,0\leq t\leq T\}\) of the critical line is \(\gg T(\ln T)^{\frac 2h} e^{-c\sqrt{\ln\ln T}}\), where \(h\) is the class number and \(c\) is an absolute constant.
See also the review of the announcement of the result [Dokl. Math. 81, No. 2, 303–308 (2010); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 431, No. 6, 741–746 (2010; Zbl 1233.11097)].

11M41 Other Dirichlet series and zeta functions
11R42 Zeta functions and \(L\)-functions of number fields
Zbl 1233.11097
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