Properties

Degree 4
Conductor $ 2^{2} \cdot 7537^{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  − 1.73·3-s − 4-s − 2.14·5-s − 0.431·7-s + 9-s + 0.986·11-s + 1.73·12-s + 0.938·13-s + 3.71·15-s + 16-s − 3.42·17-s + 0.144·19-s + 2.14·20-s + 0.748·21-s + 0.951·23-s + 1.96·25-s + 0.431·28-s + 0.845·29-s − 0.921·31-s − 1.70·33-s + 0.927·35-s − 36-s − 1.15·37-s − 1.62·39-s − 0.496·43-s − 0.986·44-s − 2.14·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{2} \cdot 7537^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{2} \cdot 7537^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{2} \cdot 7537^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1}(1-7537^{- s})^{-1}\prod_{p \nmid 30148 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.