Properties

Degree $3$
Conductor $619113924$
Sign $1$
Motivic weight $2$
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 0.5·2-s + 0.333·3-s + 0.250·4-s − 0.800·5-s + 0.166·6-s − 0.857·7-s + 0.125·8-s + 0.111·9-s − 0.400·10-s + 0.0909·11-s + 0.0833·12-s + 0.0769·13-s − 0.428·14-s − 0.266·15-s + 0.0625·16-s − 0.470·17-s + 0.0555·18-s − 0.947·19-s − 0.200·20-s − 0.285·21-s + 0.0454·22-s − 0.608·23-s + 0.0416·24-s + 1.43·25-s + 0.0384·26-s + 0.0370·27-s − 0.214·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2} \cdot 29^{2}\right)^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{2})\end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2} \cdot 29^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 2^{2} \cdot 3^{2} \cdot 11^{2} \cdot 13^{2} \cdot 29^{2} ,\ ( 1 : 1.0 ),\ 1 )\)

Particular Values

L(1/2): not computed L(1): not computed

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-11^{- s})^{-1}(1-13^{- s})^{-1}(1-29^{- s})^{-1}\prod_{p \nmid 24882 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Zeros not available.

Graph of the $Z$-function along the critical line

Plot not available.