Properties

Degree $4$
Conductor $265531392$
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 + 0.0370·9-s − 0.986·11-s + 0.938·13-s + 1.02·17-s + 1.06·19-s + 0.543·23-s − 2·25-s − 0.00712·27-s − 0.691·29-s + 0.189·33-s − 1.01·37-s − 0.180·39-s + 0.548·41-s − 0.624·43-s − 0.968·47-s − 0.197·51-s − 1.08·53-s − 0.204·57-s − 2.17·61-s + 0.860·67-s − 0.104·69-s + 0.461·71-s − 0.455·73-s + 0.384·75-s − 0.808·79-s + 0.00137·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

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

Particular Values

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

Euler product

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

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.