Properties

Degree $4$
Conductor $5184$
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  − 1.07·5-s + 0.657·11-s + 0.938·13-s + 0.856·17-s + 1.06·19-s + 1.30·23-s + 0.912·25-s + 0.845·29-s + 0.0926·31-s − 1.01·37-s − 1.05·41-s − 0.993·43-s − 2·49-s − 0.528·53-s − 0.706·55-s + 0.900·59-s + 0.495·61-s − 1.00·65-s + 0.860·67-s + 1.04·71-s − 0.737·73-s + 1.07·79-s − 0.793·83-s − 0.918·85-s − 1.01·89-s − 1.14·95-s − 0.397·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 5184,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.195883060\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.015052879\]

Euler product

\(L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 72 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line