# Super-positivity of a family of L-functions in the level aspect

Super-positivity of a family of L-functions in the level aspect An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $$s=1/2$$ s = 1 / 2 are nonnegative and all derivatives at a real point $$s > 1/2$$ s > 1 / 2 are positive. In this paper, we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight 2 and large prime level q have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $$\mathrm{Re}(s) > 0$$ Re ( s ) > 0 except possibly at $$s = 1/2.$$ s = 1 / 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# Super-positivity of a family of L-functions in the level aspect

, Volume 5 (2) – Mar 13, 2018
26 pages

/lp/springer_journal/super-positivity-of-a-family-of-l-functions-in-the-level-aspect-Ew7cuYo6Mo
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
D.O.I.
10.1007/s40687-018-0134-4
Publisher site
See Article on Publisher Site

### Abstract

An automorphic self dual L-function has the super-positivity property if all derivatives of the completed L-function at the central point $$s=1/2$$ s = 1 / 2 are nonnegative and all derivatives at a real point $$s > 1/2$$ s > 1 / 2 are positive. In this paper, we prove that at least 12% of L-functions associated to Hecke basis cusp forms of weight 2 and large prime level q have the super-positivity property. It is also shown that at least 49% of such L-functions have no real zeros on $$\mathrm{Re}(s) > 0$$ Re ( s ) > 0 except possibly at $$s = 1/2.$$ s = 1 / 2 .

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 13, 2018

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