Properties

Degree 4
Conductor 7112448
Sign $1$
Self-dual yes
Motivic weight 3

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{3}) = 1^{\mathstrut}$  − 1.073·5-s + 0.053·7-s − 0.657·11-s + 0.938·13-s + 0.171·17-s + 1.062·19-s + 0.912·25-s − 0.691·29-s − 0.057·35-s − 1.013·37-s + 0.594·41-s − 0.993·43-s + 0.002·49-s + 1.088·53-s + 0.706·55-s + 0.688·59-s + 0.495·61-s − 1.007·65-s + 0.860·67-s + 1.058·73-s − 0.035·77-s + 2.233·79-s + 0.349·83-s − 0.183·85-s + 0.300·89-s + 0.050·91-s − 1.140·95-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7112448\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 7112448,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{3}) = (1-7^{- s})^{-1}\prod_{p \nmid 1008 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 2.2905238705\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.9617090133\]

Imaginary part of the first few zeros on the critical line

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