Properties

Degree $4$
Conductor $857500$
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  − 4-s + 0.0539·7-s − 2·9-s − 0.411·11-s − 1.28·13-s + 16-s − 1.02·17-s + 0.144·19-s − 1.00·23-s − 0.0539·28-s + 0.941·29-s − 0.672·31-s + 2·36-s − 0.777·37-s + 1.00·41-s + 0.918·43-s + 0.411·44-s + 0.558·47-s + 0.00291·49-s + 1.28·52-s + 0.155·53-s − 2.40·59-s − 0.889·61-s − 0.107·63-s − 64-s − 0.683·67-s + 1.02·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 857500 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(857500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{3}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 857500,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1} (1-7^{- s})^{-1}\prod_{p \nmid 700 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.