Properties

Degree 4
Conductor $ 2^{3} \cdot 31^{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  + 0.353·2-s + 0.125·4-s + 1.07·5-s + 0.0441·8-s − 2·9-s + 0.379·10-s − 0.938·13-s + 0.0156·16-s − 0.171·17-s − 0.707·18-s − 1.06·19-s + 0.134·20-s + 1.30·23-s + 0.912·25-s − 0.331·26-s − 0.691·29-s − 0.00579·31-s + 0.00552·32-s − 0.0605·34-s − 0.250·36-s + 1.15·37-s − 0.375·38-s + 0.0474·40-s + 1.05·41-s − 0.624·43-s − 2.14·45-s + 0.461·46-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 238328 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(238328\)    =    \(2^{3} \cdot 31^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 238328,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+31^{ -s})^{-1}\prod_{p \nmid 62 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.