Properties

Degree $5$
Conductor $216090000$
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 + 3-s + 0.0625·4-s + 0.0400·5-s + 0.250·6-s + 0.0204·7-s + 0.0156·8-s + 2·9-s + 0.0100·10-s + 11-s + 0.0625·12-s + 0.171·13-s + 0.00510·14-s + 0.0400·15-s + 0.00390·16-s + 17-s + 0.5·18-s + 0.412·19-s + 0.00250·20-s + 0.0204·21-s + 0.250·22-s + 23-s + 0.0156·24-s + 0.00160·25-s + 0.0428·26-s + 2·27-s + 0.00127·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{4})=\mathstrut &\left(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\R}(s) \, \Gamma_{\C}(s+2) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{4})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{4})\end{aligned}\]

Invariants

Degree: \(5\)
Conductor: \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((5,\ 2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{4} ,\ ( 0 : 2.0, 1.0 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{4}) = (1-2^{- s})^{-1}(1-9\ 3^{- s}-81 \ 3^{-2 s}+729\ 3^{-3 s})^{-1}(1-5^{- s})^{-1}(1-7^{- s})^{-1}\prod_{p \nmid 4410 }\prod_{j=0}^{4} \left(1- \frac{\alpha_p^j\beta_p^{4-j}}{p^{s}} \right)^{-1}\)

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.