Properties

Degree $4$
Conductor $337792032000$
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 + 1.07·7-s + 0.938·13-s + 0.171·17-s − 0.543·23-s + 0.00800·25-s + 1.07·29-s + 0.0965·35-s − 1.15·37-s + 0.548·41-s + 0.581·43-s + 0.558·47-s + 0.553·49-s + 0.528·53-s + 2.17·61-s + 0.0839·65-s − 0.860·67-s − 0.0401·71-s − 1.05·73-s + 1.07·79-s − 0.428·83-s + 0.0153·85-s + 1.08·89-s + 1.01·91-s − 0.992·97-s + 0.981·101-s + 0.758·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{3} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{8} \cdot 3^{4} \cdot 5^{3} \cdot 19^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{8} \cdot 3^{4} \cdot 5^{3} \cdot 19^{4} ,\ ( \ : 1.5, 0.5 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-5^{- s})^{-1} \prod_{p \nmid 259920 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.