Properties

Degree $4$
Conductor $1.422\times 10^{12}$
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  − 0.431·7-s + 0.657·11-s + 0.938·13-s + 0.856·17-s + 0.0120·19-s − 1.30·23-s − 0.845·29-s − 1.06·31-s − 1.15·37-s − 0.594·41-s + 2.46·43-s − 0.180·49-s + 1.08·53-s − 0.461·61-s + 0.860·67-s − 1.04·71-s + 0.455·73-s − 0.284·77-s + 0.239·79-s − 1.07·83-s + 1.01·89-s − 0.405·91-s − 2.44·97-s + 1.00·101-s − 0.765·103-s − 0.758·107-s + 1.03·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 19^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-19^{- s})^{-1}\prod_{p \nmid 273600 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.