Properties

Degree $4$
Conductor $90927485064$
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  + 0.353·2-s + 0.125·4-s + 0.431·7-s + 0.0441·8-s + 2.30·11-s + 0.152·14-s + 0.0156·16-s + 0.0142·17-s − 1.06·19-s + 0.814·22-s − 2·25-s + 0.0539·28-s − 1.06·31-s + 0.00552·32-s + 0.00504·34-s − 1.03·37-s − 0.375·38-s − 1.05·41-s − 0.624·43-s + 0.287·44-s − 0.180·49-s − 0.707·50-s − 1.08·53-s + 0.0190·56-s + 0.889·61-s − 0.376·62-s + 0.00195·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{4} \cdot 13^{4} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{3} \cdot 3^{4} \cdot 13^{4} \cdot 17^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{3} \cdot 3^{4} \cdot 13^{4} \cdot 17^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1} (1-17^{- s})^{-1}\prod_{p \nmid 51714 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.