Properties

Degree $5$
Conductor $14641$
Sign $1$
Motivic weight $4$
Arithmetic yes
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 + 0.111·3-s + 4-s + 0.440·5-s − 0.111·6-s − 0.387·7-s − 8-s + 0.197·9-s − 0.440·10-s + 0.00826·11-s + 0.111·12-s − 1.17·13-s + 0.387·14-s + 0.0488·15-s + 2·16-s + 0.349·17-s − 0.197·18-s + 19-s + 0.440·20-s − 0.0430·21-s − 0.00826·22-s + 0.871·23-s − 0.111·24-s + 0.985·25-s + 1.17·26-s − 0.142·27-s − 0.387·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 14641 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(14641\)    =    \(11^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 14641,\ (0:2.0, 1.0),\ 1)\)

Particular Values

\[L(1/2, E, \mathrm{sym}^{4}) \approx 0.6058003920\] \[L(1, E, \mathrm{sym}^{4}) \approx 0.7308059068\]

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-11^{- s})^{-1}\prod_{p \nmid 11 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

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