Properties

Degree $4$
Conductor $3723875$
Sign $-1$
Motivic weight $3$
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  + 0.962·3-s − 2·4-s − 0.0894·5-s − 0.185·9-s + 0.657·11-s − 1.92·12-s − 1.28·13-s − 0.0860·15-s + 3·16-s − 0.642·17-s + 0.446·19-s + 0.178·20-s + 1.30·23-s + 0.00800·25-s − 0.285·27-s − 2.68·29-s − 0.00579·31-s + 0.633·33-s + 0.370·36-s − 0.324·37-s − 1.23·39-s + 0.834·41-s + 0.918·43-s − 1.31·44-s + 0.0165·45-s + 1.08·47-s + 2.88·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 3723875 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3723875\)    =    \(5^{3} \cdot 31^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 3723875,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+5^{ -s})^{-1}(1+31^{ -s})^{-1}\prod_{p \nmid 155 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.