Properties

Degree 4
Conductor $ 2^{10} \cdot 3^{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 − 1.07·5-s − 0.431·7-s + 0.0370·9-s − 0.657·11-s + 0.938·13-s − 0.206·15-s − 0.171·17-s + 1.06·19-s − 0.0831·21-s + 0.912·25-s + 0.00712·27-s − 0.691·29-s − 1.06·31-s − 0.126·33-s + 0.463·35-s + 0.622·37-s + 0.180·39-s − 0.594·41-s − 0.993·43-s − 0.0397·45-s − 0.744·47-s − 0.180·49-s − 0.0329·51-s − 0.155·53-s + 0.706·55-s + 0.204·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 27648 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(27648\)    =    \(2^{10} \cdot 3^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 27648,\ (\ :1.5, 0.5),\ -1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}\prod_{p \nmid 96 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.