Properties

Degree 4
Conductor $ 2^{3} \cdot 23^{3} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  − 0.353·2-s + 0.125·4-s + 2.14·5-s − 0.431·7-s − 0.0441·8-s − 2·9-s − 0.758·10-s − 0.986·11-s + 0.938·13-s + 0.152·14-s + 0.0156·16-s + 0.856·17-s + 0.707·18-s + 0.821·19-s + 0.268·20-s + 0.348·22-s + 0.00906·23-s + 1.96·25-s − 0.331·26-s − 0.0539·28-s − 0.691·29-s − 0.00552·32-s − 0.302·34-s − 0.927·35-s − 0.250·36-s + 1.03·37-s − 0.290·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 97336 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(97336\)    =    \(2^{3} \cdot 23^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 97336,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1-23^{- s})^{-1}\prod_{p \nmid 46 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.