Properties

Degree $4$
Conductor $5419008$
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  − 0.192·3-s − 1.07·5-s − 1.07·7-s + 0.0370·9-s − 0.986·11-s + 0.533·13-s + 0.206·15-s + 0.446·19-s + 0.207·21-s + 0.912·25-s − 0.00712·27-s + 1.07·29-s + 0.990·31-s + 0.189·33-s + 1.15·35-s + 0.866·37-s − 0.102·39-s − 0.685·41-s + 1.08·43-s − 0.0397·45-s + 1.08·47-s + 0.166·49-s − 1.18·53-s + 1.05·55-s − 0.0859·57-s + 0.688·59-s + 0.461·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5419008 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5419008\)    =    \(2^{12} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 5419008,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1+20\ 7^{- s}+343\ 7^{-2 s})^{-1}\prod_{p \nmid 9408 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.