## Experimental Mathematics

- Experiment. Math.
- Volume 13, Issue 2 (2004), 185-198.

### On the Vanishing of Twisted *L*-Functions of Elliptic Curves

Chantal David, Jack Fearnley, and Hershy Kisilevsky

#### Abstract

Let *E* be an elliptic curve over *q* with *L*-function {\small $L_E(s)$}. We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted *L*-functions {\small $L_E(1, \chi)$}, as {\small $\chi$} runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than *X* for which {\small $L_E(1, \chi)$} vanishes is asymptotic to {\small $b_E X^{1/2}
\log^{e_E}{X}$} for some constants {\small $b_E, e_E$} depending only on *E*. We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values {\small $L_E(1, \chi)$} in the family of cubic twists. Finally, we present
experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of {\small $L_E(s)$}.

#### Article information

**Source**

Experiment. Math., Volume 13, Issue 2 (2004), 185-198.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.em/1090350933

**Mathematical Reviews number (MathSciNet)**

MR2068892

**Zentralblatt MATH identifier**

1115.11033

**Subjects**

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

**Keywords**

Elliptic curves L-functions random matrix theory

#### Citation

David, Chantal; Fearnley, Jack; Kisilevsky, Hershy. On the Vanishing of Twisted L -Functions of Elliptic Curves. Experiment. Math. 13 (2004), no. 2, 185--198. https://projecteuclid.org/euclid.em/1090350933