Properties

Degree 4
Conductor $ 2^{6} \cdot 7^{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  − 1.07·5-s − 0.0539·7-s − 2·9-s + 0.657·11-s − 0.938·13-s − 0.171·17-s + 2.51·19-s + 0.912·25-s − 0.845·29-s + 0.0926·31-s + 0.0579·35-s + 0.622·37-s − 0.594·41-s + 0.993·43-s + 2.14·45-s + 0.744·47-s + 0.00291·49-s − 1.08·53-s − 0.706·55-s + 1.08·61-s + 0.107·63-s + 1.00·65-s + 0.860·67-s + 1.04·71-s − 0.737·73-s − 0.0355·77-s + 2.23·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 21952 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(21952\)    =    \(2^{6} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 21952,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+7^{ -s})^{-1}\prod_{p \nmid 56 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.