Properties

Degree $4$
Conductor $12005000$
Sign $-1$
Motivic weight $3$
Arithmetic yes
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 − 0.0441·8-s − 2·9-s − 0.657·11-s − 1.28·13-s + 0.0156·16-s − 0.856·17-s + 0.707·18-s + 0.232·22-s + 0.452·26-s − 0.845·29-s − 0.0926·31-s − 0.00552·32-s + 0.302·34-s − 0.250·36-s + 1.15·37-s + 0.594·41-s + 0.993·43-s − 0.0822·44-s − 0.744·47-s − 0.160·52-s − 0.528·53-s + 0.298·58-s − 0.953·59-s + 2.17·61-s + 0.0327·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 12005000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12005000\)    =    \(2^{3} \cdot 5^{4} \cdot 7^{4}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 12005000,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1} \prod_{p \nmid 2450 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.