Properties

Degree 4
Conductor $ 3^{4} \cdot 11^{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  − 1.06·2-s + 0.375·4-s − 1.07·5-s − 0.431·7-s − 0.662·8-s + 1.13·10-s − 0.938·13-s + 0.458·14-s + 0.546·16-s + 0.856·17-s − 0.402·20-s − 1.30·23-s + 0.912·25-s + 0.995·26-s − 0.161·28-s + 0.845·29-s − 0.0926·31-s + 0.580·32-s − 0.907·34-s + 0.463·35-s − 1.01·37-s + 0.711·40-s + 0.594·41-s + 1.38·46-s + 0.744·47-s − 0.180·49-s − 0.967·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(1185921\)    =    \(3^{4} \cdot 11^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 1185921,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 1089 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.