Properties

Degree 4
Conductor $ 179^{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  − 0.268·5-s − 0.431·7-s − 2·9-s − 0.657·11-s + 0.533·13-s − 16-s − 0.470·17-s + 1.05·19-s − 0.543·23-s + 0.232·25-s − 0.941·29-s − 0.0926·31-s + 0.115·35-s − 0.622·37-s + 2.83·41-s − 1.36·43-s + 0.536·45-s − 0.288·47-s − 0.180·49-s + 0.176·55-s + 1.02·59-s + 2.17·61-s + 0.863·63-s − 0.143·65-s + 0.869·67-s − 0.737·73-s + 0.284·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5735339 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5735339\)    =    \(179^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 5735339,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-179^{- s})^{-1}\prod_{p \nmid 179 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.