Properties

Degree $4$
Conductor $23708160000$
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  − 0.192·3-s − 0.0539·7-s + 0.0370·9-s + 0.657·11-s + 0.938·13-s + 0.856·17-s − 1.06·19-s + 0.0103·21-s + 1.30·23-s − 0.00712·27-s + 0.845·29-s − 0.0926·31-s − 0.126·33-s + 0.622·37-s − 0.180·39-s − 0.594·41-s − 2.46·43-s − 0.744·47-s + 0.00291·49-s − 0.164·51-s − 1.08·53-s + 0.204·57-s + 0.900·59-s − 0.495·61-s − 0.00199·63-s + 0.218·67-s − 0.251·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 7^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1} (1+7^{ -s})^{-1}\prod_{p \nmid 33600 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.