Abstract
Let f and g be two holomorphic cuspidal Hecke eigenforms on the full modular group \( \text {SL}_{2}({\mathbb {Z}}). \) We show that the Rankin–Selberg L-function \(L(s, f \times g)\) has no pole at \(s=1\) unless \( f=g \), in which case it has a pole with residue \( \frac{3}{\pi }\frac{(4\pi )^{k}}{\Gamma (k)} \Vert f \Vert ^2 \), where \( \Vert f\Vert \) is the Petersson norm of f. Our proof uses the Petersson trace formula and avoids the Rankin–Selberg method.
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Acknowledgements
The authors thank Farrell Brumley, Subhajit Jana, M. Ram Murty, V. Kumar Murty, Dipendra Prasad, Olivier Ramar and Peter Sarnak for their interest in this paper and their valuable comments. We especially thank the anonymous referee for helpful remarks which have improved the scope of the result in this paper.
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Ganguly, S., Mawia, R. Rankin–Selberg L-functions and “beyond endoscopy”. Math. Z. 296, 175–184 (2020). https://doi.org/10.1007/s00209-019-02431-5
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DOI: https://doi.org/10.1007/s00209-019-02431-5