Properties

Degree $4$
Conductor $6.253\times 10^{13}$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.962·3-s − 0.804·5-s + 0.0539·7-s − 0.185·9-s − 0.986·11-s − 0.853·13-s − 0.774·15-s − 0.171·17-s − 0.0120·19-s + 0.0519·21-s + 0.543·23-s − 0.792·25-s − 0.285·27-s + 0.403·29-s + 0.672·31-s − 0.949·33-s − 0.0434·35-s − 0.622·37-s − 0.821·39-s − 1.05·41-s − 1.06·43-s + 0.149·45-s + 0.00310·47-s + 0.00291·49-s − 0.164·51-s + 0.794·55-s − 0.0116·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{8} \cdot 7^{3} \cdot 19^{3} \cdot 47^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{8} \cdot 7^{3} \cdot 19^{3} \cdot 47^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{8} \cdot 7^{3} \cdot 19^{3} \cdot 47^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-7^{- s})^{-1}(1+19^{ -s})^{-1}(1-47^{- s})^{-1}\prod_{p \nmid 100016 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.