Properties

Degree 4
Conductor $ 2^{2} \cdot 11^{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 − 4-s + 0.268·5-s − 1.07·7-s − 0.185·9-s − 0.0274·11-s + 0.962·12-s + 0.853·13-s − 0.258·15-s + 16-s + 0.171·17-s + 2.51·19-s − 0.268·20-s + 1.03·21-s + 1.00·23-s + 0.232·25-s + 0.285·27-s + 1.07·28-s − 1.07·31-s + 0.0263·33-s − 0.289·35-s + 0.185·36-s + 0.324·37-s − 0.821·39-s − 0.496·43-s + 0.0274·44-s − 0.0496·45-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5324 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5324\)    =    \(2^{2} \cdot 11^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 5324,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1}(1+11^{ -s})^{-1}\prod_{p \nmid 44 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0.6632836559\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.6379183409\]

Imaginary part of the first few zeros on the critical line

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