Properties

Degree 4
Conductor $ 3^{3} \cdot 19^{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  − 0.192·3-s + 0.268·5-s − 2.96·7-s + 0.0370·9-s − 0.575·11-s − 0.938·13-s − 0.0516·15-s − 16-s + 0.470·17-s − 0.0120·19-s + 0.571·21-s + 1.08·23-s + 0.232·25-s − 0.00712·27-s + 0.691·29-s + 0.903·31-s + 0.110·33-s − 0.796·35-s + 0.180·39-s + 0.301·43-s + 0.00993·45-s + 0.363·47-s + 0.192·48-s + 4.77·49-s − 0.0906·51-s − 0.155·53-s − 0.154·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(185193\)    =    \(3^{3} \cdot 19^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 185193,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1+19^{ -s})^{-1}\prod_{p \nmid 57 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.