Properties

Degree 4
Conductor $ 11^{3} \cdot 13^{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  + 0.962·3-s − 2·4-s + 0.804·5-s + 1.07·7-s − 0.185·9-s − 0.0274·11-s − 1.92·12-s − 0.0213·13-s + 0.774·15-s + 3·16-s + 1.02·17-s − 0.821·19-s − 1.60·20-s + 1.03·21-s + 0.190·23-s − 0.792·25-s − 0.285·27-s − 2.15·28-s + 0.691·29-s + 0.921·31-s − 0.0263·33-s + 0.869·35-s + 0.370·36-s − 2.29·37-s − 0.0205·39-s + 0.685·41-s + 0.993·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2924207 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2924207\)    =    \(11^{3} \cdot 13^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2924207,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+11^{ -s})^{-1}(1+13^{ -s})^{-1}\prod_{p \nmid 143 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.