Properties

Degree 4
Conductor $ 3^{3} \cdot 47^{3} $
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 − 2·4-s + 0.804·5-s + 0.809·7-s + 0.0370·9-s + 1.06·11-s + 0.384·12-s + 0.853·13-s − 0.154·15-s + 3·16-s + 3.42·17-s + 0.144·19-s − 1.60·20-s − 0.155·21-s − 1.00·23-s − 0.792·25-s − 0.00712·27-s − 1.61·28-s + 0.364·29-s − 1.06·31-s − 0.205·33-s + 0.651·35-s − 0.0740·36-s − 0.324·37-s − 0.164·39-s − 0.685·41-s + 0.624·43-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

\( d \)  =  \(4\)
\( N \)  =  \(2803221\)    =    \(3^{3} \cdot 47^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2803221,\ (\ :1.5, 0.5),\ -1)$

Euler product

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

Particular Values

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

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.