Properties

Degree $4$
Conductor $7112448$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  − 1.07·5-s − 0.0539·7-s − 2.30·11-s − 1.28·13-s + 0.856·17-s + 1.06·19-s + 0.761·23-s + 0.912·25-s + 0.307·29-s + 1.06·31-s + 0.0579·35-s + 1.01·37-s + 0.685·41-s − 0.993·43-s − 0.968·47-s + 0.00291·49-s + 0.933·53-s + 2.47·55-s + 0.688·59-s + 0.495·61-s + 1.37·65-s − 0.218·67-s + 1.06·71-s + 0.455·73-s + 0.124·77-s − 1.07·79-s − 0.918·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7112448\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 7112448,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}\prod_{p \nmid 1008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.