Properties

Degree $5$
Conductor $1336336$
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  + 0.250·2-s − 1.22·3-s + 0.0625·4-s + 5-s − 0.305·6-s − 0.632·7-s + 0.0156·8-s + 0.679·9-s + 0.250·10-s + 1.89·11-s − 0.0763·12-s + 0.171·13-s − 0.158·14-s − 1.22·15-s + 0.00390·16-s + 0.00346·17-s + 0.169·18-s − 0.817·19-s + 0.0625·20-s + 0.773·21-s + 0.473·22-s + 23-s − 0.0190·24-s + 3·25-s + 0.0428·26-s + 0.980·27-s − 0.0395·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut & 1336336 ^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{4}) \end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(1336336\)    =    \(2^{4} \cdot 17^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 1336336,\ (0:2.0, 1.0),\ 1)\)

Particular Values

L(1/2): not computed L(1): not computed

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1+11\ 3^{- s}+66\ 3^{-2 s}-594 \ 3^{-3 s}-8019 \ 3^{-4 s}-59049 \ 3^{-5 s})^{-1}(1-29\ 13^{- s}-8294 \ 13^{-2 s}+1401686\ 13^{-3 s}+139977461\ 13^{-4 s}-137858491849 \ 13^{-5 s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 51714 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

Imaginary part of the first few zeros on the critical line

Zeros not available.

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

Plot not available.