Properties

Degree $4$
Conductor $160699616000$
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  − 0.0894·5-s − 0.0539·7-s − 2·9-s + 1.28·13-s + 0.856·17-s + 0.00800·25-s + 0.845·29-s − 0.0926·31-s + 0.00482·35-s − 1.15·37-s + 0.594·41-s − 0.993·43-s + 0.178·45-s + 0.744·47-s + 0.00291·49-s + 0.528·53-s − 0.953·59-s + 2.17·61-s + 0.107·63-s − 0.114·65-s + 0.218·67-s + 3.04·71-s + 0.455·73-s + 1.07·79-s + 3·81-s − 1.07·83-s − 0.0765·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{8} \cdot 5^{3} \cdot 7^{3} \cdot 11^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+5^{ -s})^{-1}(1+7^{ -s})^{-1} \prod_{p \nmid 67760 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.