Properties

Degree 4
Conductor 8000
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  + 0.089·5-s − 0.431·7-s − 2·9-s − 0.657·11-s + 0.938·13-s − 0.856·17-s − 1.062·19-s − 1.087·23-s + 0.008·25-s + 0.691·29-s − 0.092·31-s − 0.038·35-s − 1.013·37-s + 1.051·41-s + 0.624·43-s − 0.178·45-s − 0.968·47-s − 0.180·49-s − 1.088·53-s − 0.058·55-s + 0.900·59-s + 0.495·61-s + 0.863·63-s + 0.083·65-s − 1.021·67-s + 1.058·73-s + 0.284·77-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 8000 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & - \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(8000\)    =    \(2^{6} \cdot 5^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 8000,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{3}) = (1-5^{- s})^{-1}\prod_{p \nmid 40 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{equation}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3})=0\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.6440527445\]

Imaginary part of the first few zeros on the critical line

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