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  + 0.250·2-s + 0.111·3-s + 0.0625·4-s + 5-s + 0.0277·6-s − 0.387·7-s + 0.0156·8-s + 0.197·9-s + 0.250·10-s − 0.785·11-s + 0.00694·12-s − 1.17·13-s − 0.0969·14-s + 0.111·15-s + 0.00390·16-s − 0.307·17-s + 0.0493·18-s − 1.21·19-s + 0.0625·20-s − 0.0430·21-s − 0.196·22-s − 1.24·23-s + 0.00173·24-s + 25-s − 0.294·26-s − 0.142·27-s − 0.0242·28-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

\[\begin{aligned}L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-25\ 5^{- s})^{-1}\prod_{p \nmid 50 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{4}) \approx 0.9208160997\] \[L(1, E, \mathrm{sym}^{4}) \approx 1.141876031\]

Imaginary part of the first few zeros on the critical line

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