Properties

Degree $4$
Conductor $2803221$
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  + 0.192·3-s + 0.804·5-s + 0.809·7-s + 0.0370·9-s − 0.575·11-s + 0.938·13-s + 0.154·15-s − 16-s − 0.856·17-s − 0.144·19-s + 0.155·21-s − 1.00·23-s − 0.792·25-s + 0.00712·27-s − 0.941·29-s − 0.672·31-s − 0.110·33-s + 0.651·35-s + 0.777·37-s + 0.180·39-s + 0.685·41-s − 0.496·43-s + 0.0298·45-s − 0.00310·47-s − 0.192·48-s + 0.860·49-s − 0.164·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2803221 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2803221\)    =    \(3^{3} \cdot 47^{3}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2803221,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+47^{ -s})^{-1}\prod_{p \nmid 141 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.