Properties

Degree 4
Conductor $ 2^{3} \cdot 3^{4} $
Sign $1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.353·2-s + 0.125·4-s − 0.268·5-s − 0.431·7-s + 0.0441·8-s − 9-s − 0.0948·10-s + 0.533·13-s − 0.152·14-s + 0.0156·16-s − 1.07·17-s − 0.353·18-s + 1.06·19-s − 0.0335·20-s + 0.232·25-s + 0.188·26-s − 0.0539·28-s − 1.32·29-s + 1.06·31-s + 0.00552·32-s − 0.378·34-s + 0.115·35-s − 0.125·36-s + 0.324·37-s + 0.375·38-s − 0.0118·40-s + 1.05·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(648\)    =    \(2^{3} \cdot 3^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((4,\ 648,\ (\ :1.5, 0.5),\ 1)\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+27\ 3^{-2 s})^{-1}\prod_{p \nmid 162 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0.8470172594\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.011529058\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line