Properties

Degree 4
Conductor $ 2^{3} \cdot 5^{4} $
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  + 0.353·2-s + 0.962·3-s + 0.125·4-s + 0.340·6-s + 1.07·7-s + 0.0441·8-s − 0.185·9-s + 1.06·11-s + 0.120·12-s − 0.853·13-s + 0.381·14-s + 0.0156·16-s − 1.07·17-s − 0.0654·18-s − 0.784·19-s + 1.03·21-s + 0.377·22-s + 0.543·23-s + 0.0425·24-s − 0.301·26-s − 0.285·27-s + 0.134·28-s − 0.672·31-s + 0.00552·32-s + 1.02·33-s − 0.378·34-s − 0.0231·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 5000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(5000\)    =    \(2^{3} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 5000,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1} \prod_{p \nmid 50 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 2.169371233\] \[L(1, E, \mathrm{sym}^{3}) \approx 1.713460656\]

Imaginary part of the first few zeros on the critical line

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