Properties

Degree $3$
Conductor $28224$
Sign $1$
Motivic weight $2$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  + 0.333·3-s − 5-s + 0.142·7-s + 0.111·9-s − 0.636·11-s − 0.692·13-s − 0.333·15-s − 0.0588·17-s − 0.157·19-s + 0.0476·21-s + 0.565·23-s + 2·25-s + 0.0370·27-s − 0.862·29-s − 31-s − 0.212·33-s − 0.142·35-s − 0.0270·37-s − 0.230·39-s + 0.560·41-s + 0.488·43-s − 0.111·45-s − 0.659·47-s + 0.0204·49-s − 0.0196·51-s − 0.320·53-s + 0.636·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 28224 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned}\]

Invariants

Degree: \(3\)
Conductor: \(28224\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((3,\ 28224,\ (1:1.0),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{2}) = (1-3^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 9408 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\)

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.