Properties

Degree 4
Conductor $ 2^{6} \cdot 3^{4} \cdot 139^{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 + 0.809·7-s − 1.06·11-s + 1.08·13-s + 0.171·17-s − 0.821·19-s + 1.30·23-s + 0.232·25-s − 1.05·29-s + 0.353·31-s − 0.217·35-s − 1.15·37-s + 0.594·41-s − 1.06·43-s − 0.744·47-s + 0.860·49-s − 0.933·53-s + 0.286·55-s − 0.503·59-s + 0.495·61-s − 0.291·65-s − 0.260·67-s − 0.666·71-s + 1.05·73-s − 0.865·77-s − 1.43·79-s + 0.654·83-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{6} \cdot 3^{4} \cdot 139^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{6} \cdot 3^{4} \cdot 139^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{6} \cdot 3^{4} \cdot 139^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+139^{ -s})^{-1}\prod_{p \nmid 10008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.