Properties

Degree $4$
Conductor $31116960000$
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  − 0.986·11-s − 0.938·13-s − 1.02·17-s − 1.30·23-s + 0.672·31-s + 0.355·37-s + 0.594·41-s + 0.581·43-s + 0.186·47-s + 0.528·53-s + 0.900·59-s − 0.461·61-s − 0.474·67-s − 0.0401·71-s − 0.737·73-s − 2.23·79-s + 1.90·83-s + 0.300·89-s − 0.992·97-s + 0.0827·101-s + 3.71·103-s + 0.758·107-s + 0.376·109-s + 0.349·113-s − 0.0676·121-s + 0.922·127-s + 0.944·131-s + ⋯

Functional equation

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

Invariants

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

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 176400 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.