Properties

Degree $4$
Conductor $6615000$
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  − 0.353·2-s − 0.192·3-s + 0.125·4-s + 0.0680·6-s + 1.07·7-s − 0.0441·8-s + 0.0370·9-s + 0.575·11-s − 0.0240·12-s − 3.43·13-s − 0.381·14-s + 0.0156·16-s − 1.02·17-s − 0.0130·18-s − 0.446·19-s − 0.207·21-s − 0.203·22-s + 0.407·23-s + 0.00850·24-s + 1.21·26-s − 0.00712·27-s + 0.134·28-s − 0.307·29-s − 0.903·31-s − 0.00552·32-s − 0.110·33-s + 0.363·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 6615000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6615000\)    =    \(2^{3} \cdot 3^{3} \cdot 5^{4} \cdot 7^{2}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 6615000,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1}(1+3^{ -s})^{-1} (1-20\ 7^{- s}+343\ 7^{-2 s})^{-1}\prod_{p \nmid 7350 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.