Properties

Degree $3$
Conductor $24025$
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  − 2-s − 0.666·3-s + 2·4-s + 0.200·5-s + 0.666·6-s − 7-s − 2·8-s + 1.11·9-s − 0.200·10-s + 0.454·11-s − 1.33·12-s + 1.76·13-s + 14-s − 0.133·15-s + 3·16-s + 0.470·17-s − 1.11·18-s − 0.947·19-s + 0.400·20-s + 0.666·21-s − 0.454·22-s + 1.78·23-s + 1.33·24-s + 0.0400·25-s − 1.76·26-s − 0.185·27-s − 2·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 24025 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(24025\)    =    \(5^{2} \cdot 31^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 24025,\ (1:1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-5^{- s})^{-1}(1-31^{- s})^{-1}\prod_{p \nmid 155 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

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.