Properties

Degree $4$
Conductor $840983040000$
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  + 0.192·3-s + 0.0370·9-s + 0.657·11-s + 0.938·13-s + 0.171·17-s + 1.06·19-s + 0.00906·23-s + 0.00712·27-s − 0.691·29-s + 0.126·33-s + 0.622·37-s + 0.180·39-s + 0.685·41-s + 0.993·43-s − 2·49-s + 0.0329·51-s − 1.08·53-s + 0.204·57-s − 0.900·59-s − 0.461·61-s − 0.218·67-s + 0.00174·69-s + 1.04·71-s + 0.737·73-s + 1.07·79-s + 0.00137·81-s + 0.793·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 23^{3}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{12} \cdot 3^{3} \cdot 5^{4} \cdot 23^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1} (1-23^{- s})^{-1}\prod_{p \nmid 110400 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.