Properties

Degree 4
Conductor $ 3^{4} \cdot 7^{3} \cdot 11^{2} $
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 + 0.0539·7-s + 0.938·13-s − 16-s − 1.07·17-s − 1.06·19-s + 0.761·23-s + 0.232·25-s + 0.307·29-s − 0.672·31-s − 0.0144·35-s + 1.15·37-s − 0.594·41-s + 0.159·43-s + 0.363·47-s + 0.00291·49-s − 0.870·53-s − 0.721·59-s − 0.461·61-s − 0.251·65-s + 0.829·67-s − 1.26·71-s − 0.737·73-s − 0.808·79-s + 0.268·80-s − 0.218·83-s + 0.287·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3361743 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3361743\)    =    \(3^{4} \cdot 7^{3} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3361743,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-7^{- s})^{-1}(1+1331\ 11^{-2 s})^{-1}\prod_{p \nmid 7623 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.