Remarks on the rightmost critical value of the triple product \(L\)-function. (English) Zbl 1499.11205
Summary: Let \(f\in \mathcal{S}_k(\mathrm{SL}_2(\mathbb Z))\) be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product \(L\)-functions \(L(s,f\otimes g\otimes h)\), where \(g\) and \(h\) run over an orthogonal basis of \(\mathcal{S}_k(\mathrm{SL}_2(\mathbb Z))\) consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product \(L\)-functions.
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
| 11F30 | Fourier coefficients of automorphic forms |
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