Properties

Degree 5
Conductor $ 2^{4} \cdot 5^{4} $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{4})$  = 1  + 2-s − 1.22·3-s + 4-s + 0.0400·5-s − 1.22·6-s − 0.387·7-s + 8-s + 0.679·9-s + 0.0400·10-s + 11-s − 1.22·12-s + 0.171·13-s − 0.387·14-s − 0.0488·15-s + 16-s − 0.868·17-s + 0.679·18-s − 0.817·19-s + 0.0400·20-s + 0.473·21-s + 22-s − 1.24·23-s − 1.22·24-s + 0.00160·25-s + 0.171·26-s + 0.980·27-s − 0.387·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 10000 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

\( d \)  =  \(5\)
\( N \)  =  \(10000\)    =    \(2^{4} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(5,\ 10000,\ (0:2.0, 1.0),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-4\ 2^{- s})^{-1}(1-5^{- s})^{-1}\prod_{p \nmid 100 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{4}) \approx 0.9478365060\] \[L(1, E, \mathrm{sym}^{4}) \approx 1.181662623\]

Imaginary part of the first few zeros on the critical line

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