Properties

Degree 4
Conductor $ 3^{3} \cdot 11^{4} \cdot 13^{4} $
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  − 0.192·3-s − 2.14·5-s − 0.701·7-s + 0.0370·9-s + 0.413·15-s − 16-s + 1.02·17-s + 1.05·19-s + 0.135·21-s − 0.761·23-s + 1.96·25-s − 0.00712·27-s + 0.845·29-s − 1.07·31-s + 1.50·35-s + 0.866·37-s + 0.594·41-s − 2.46·43-s − 0.0795·45-s + 0.558·47-s + 0.192·48-s − 1.09·49-s − 0.197·51-s − 1.08·53-s − 0.202·57-s − 0.397·59-s + 0.711·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{3} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3^{3} \cdot 11^{4} \cdot 13^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 3^{3} \cdot 11^{4} \cdot 13^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1} \prod_{p \nmid 61347 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.