Properties

Degree 4
Conductor $ 2^{3} \cdot 3^{3} \cdot 29^{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 + 0.192·3-s + 0.125·4-s + 0.804·5-s + 0.0680·6-s − 0.701·7-s + 0.0441·8-s + 0.0370·9-s + 0.284·10-s + 0.986·11-s + 0.0240·12-s − 0.248·14-s + 0.154·15-s + 0.0156·16-s + 1.07·17-s + 0.0130·18-s + 0.446·19-s + 0.100·20-s − 0.135·21-s + 0.348·22-s + 1.08·23-s + 0.00850·24-s − 0.792·25-s + 0.00712·27-s − 0.0877·28-s + 0.00640·29-s + 0.0547·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5268024 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5268024\)    =    \(2^{3} \cdot 3^{3} \cdot 29^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 5268024,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-3^{- s})^{-1}(1-29^{- s})^{-1}\prod_{p \nmid 174 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.