Moments of L′(½) in the Family of Quadratic Twists

Ian Petrow

Type: Article

Publication Date: 2012-12-16

Citations: 12

DOI: https://doi.org/10.1093/imrn/rns265

Abstract

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to −1 and odd orthogonal symmetry. Assuming generalized Riemann hypothesis, we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g, and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves using the modularity theorem and the work of Gross and Zagier.

Locations

International Mathematics Research Notices - View
UCL Discovery (University College London) - View - PDF
arXiv (Cornell University) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+	Moments of quadratic twists of modular $L$-functions	2022	Xiannan Li
+ PDF Chat	The second moment of quadratic twists of modular $L$-functions	2010	K. Soundararajan Matthew P. Young
+ PDF Chat	Lower bounds for moments of quadratic twists of modular $L$-functions	2024	Peng Gao Liangyi Zhao
+ PDF Chat	Sharp lower bounds for negative moments of quadratic twist of modular L-functions	2023	Huaqing Bian Xiaofei Yan Ruiyang Yue
+ PDF Chat	The first moment of quadratic twists of modular $L$-functions	2022	Quanli Shen
+	Lower bounds for negative moments of quadratic twist of modular L-functions	2024	Huaqing Bian Xiaofei Yan Ruiyang Yue
+	Moments of quadratic twists of modular $L$-functions	2024	Xiannan Li
+	Lower bounds for moments of quadratic twists of modular $L$-functions	2021	Peng Gao Liangyi Zhao
+ PDF Chat	Moment of derivatives of L-functions for two distinct newforms	2024	Se Woon Choi B. Kim Hansol Kim H.-M. Kim Wonwoong Lee
+	The first moment of quadratic twists of modular $L$-functions	2021	Quanli Shen
+	Secondary terms in the number of vanishings of quadratic twists of elliptic curve L-functions	2005	J. Brian Conrey Atul Pocharel Michael Rubinstein Mark E. Watkins
+ PDF Chat	Bounds for Moments of Twisted Fourier coefficients of Modular Forms	2024	Peng Gao Liangyi Zhao
+	Lower order terms for the moments of symplectic and orthogonal families of $L$-functions	2012	I. P. Goulden Duc Khiem Huynh Rishikesh Michael Rubinstein
+	Lower order terms for the moments of symplectic and orthogonal families of $L$-functions	2012	I. P. Goulden Duc Khiem Huynh Rishikesh Michael Rubinstein
+ PDF Chat	Lower order terms for the moments of symplectic and orthogonal families of L -functions	2012	I. P. Goulden Duc Khiem Huynh Rishikesh Michael Rubinstein
+ PDF Chat	A uniform asymptotic formula for the second moment of primitive L-functions on the critical line	2016	Olga Balkanova Dmitry Frolenkov
+	Moments of L-functions attached to the twist of modular form by Dirichlet characters	2015	Guanghua Ji Haiwei Sun
+	Moments of L-Functions Attached to the Twist of Modular Form by Dirichlet Characters	2015	Guanghua Ji Haiwei Sun
+	Moments of the products of quadratic twists of automorphic L-functions	2016	Keiju Sono
+	Moments of derivatives of modular $L$-functions	2023	Sumit Kumar Kummari Mallesham Prahlad Sharma Saurabh Kumar Singh

Works That Cite This (12)

Action	Title	Year	Authors
+	The first moment of quadratic twists of modular $L$-functions	2021	Quanli Shen
+	Moments of quadratic twists of modular $L$-functions	2024	Xiannan Li
+	Moments of Dirichlet L–functions with prime conductors over function fields	2020	Hung M. Bui Alexandra Florea
+	Lower bounds for moments of quadratic twists of modular $L$-functions	2021	Peng Gao Liangyi Zhao
+ PDF Chat	Lower bounds for moments of quadratic twists of modular $L$-functions	2024	Peng Gao Liangyi Zhao
+ PDF Chat	Moments of quadratic twists of elliptic curve L-functions over function fields	2020	Hung M. Bui Alexandra Florea Jonathan P. Keating Edva Roditty-Gershon
+ PDF Chat	Rational Points on Elliptic K3 Surfaces of Quadratic Twist Type	2020	Zhizhong Huang
+	Moments of Derivatives of Modular <i>L</i>-Functions	2024	Sumit Kumar Kummari Mallesham Prahlad Sharma Saurabh Kumar Singh
+	Moments and Non-vanishing of central values of Quadratic Hecke $L$-functions in the Gaussian Field	2020	Peng Gao
+ PDF Chat	First moments of some Hecke $L$-functions of prime moduli	2022	Peng Gao Liangyi Zhao

Works Cited by This (25)

Action	Title	Year	Authors
+	The first non-vanishing quadratic twist of an automorphic L-series	2010	Jeffrey Hoffstein Alex Kontorovich
+	On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises	2001	Christophe Breuil Brian Conrad Fred Diamond Richard Taylor
+ PDF Chat	Moments of the Riemann zeta function	2009	K. Soundararajan
+	Nonvanishing theorems for L-functions of modular forms and their derivatives	1990	Daniel Bump Solomon Friedberg Jeffrey Hoffstein
+ PDF Chat	On mean values and non-vanishing of derivatives of <i>L</i>-functions in a nonlinear family	2010	Ritabrata Munshi
+ PDF Chat	The Average Analytic Rank of Elliptic Curves	2004	D. R. Heath‐Brown
+	Mean Values of Derivatives of Modular L-Series	1991	M. Ram Murty V. Kumar Murty
+ PDF Chat	On the distribution of the nontrivial zeros of quadratic L-functions of imaginary quadratic number fields close to the real axis	2006	Ali E. Özlük C. Snyder
+ PDF Chat	On the order of vanishing of modular $L$-functions at the critical point	1990	Henryk Iwaniec
+ PDF Chat	On quadratic families of CM elliptic curves	2011	Ritabrata Munshi