Properties

Degree $4$
Conductor $343000$
Sign $1$
Motivic weight $3$
Arithmetic yes
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.353·2-s + 0.125·4-s − 0.0894·5-s − 0.0539·7-s + 0.0441·8-s − 2·9-s − 0.0316·10-s − 0.657·11-s − 1.28·13-s − 0.0190·14-s + 0.0156·16-s − 0.856·17-s − 0.707·18-s − 0.0111·20-s − 0.232·22-s + 0.00800·25-s − 0.452·26-s − 0.00674·28-s − 0.845·29-s + 0.0926·31-s + 0.00552·32-s − 0.302·34-s + 0.00482·35-s − 0.250·36-s − 1.15·37-s − 0.00395·40-s − 0.594·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 343000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(343000\)    =    \(2^{3} \cdot 5^{3} \cdot 7^{3}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 343000,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+5^{ -s})^{-1}(1+7^{ -s})^{-1}\prod_{p \nmid 70 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.