Journal of the European Mathematical Society

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Volume 18, Issue 6, 2016, pp. 1349–1389
DOI: 10.4171/JEMS/616

Twists and resonance of $L$-functions, I

Jerzy Kaczorowski[1] and Alberto Perelli[2]

(1) Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Umultowska 87, 61-614, Poznan, Poland
(2) DIMA - Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146, Genova, Italy

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $≤ 1 / d$ of the $L$-functions of any degree $d ≥ 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

Keywords: $L$-functions, Selberg class, twists, resonance

Kaczorowski Jerzy, Perelli Alberto: Twists and resonance of $L$-functions, I. J. Eur. Math. Soc. 18 (2016), 1349-1389. doi: 10.4171/JEMS/616