Properties

Degree $4$
Conductor $132651000$
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 + 0.0894·5-s − 0.0680·6-s + 0.0441·8-s + 0.0370·9-s + 0.0316·10-s − 0.657·11-s − 0.0240·12-s + 0.938·13-s − 0.0172·15-s + 0.0156·16-s + 0.0142·17-s + 0.0130·18-s − 1.06·19-s + 0.0111·20-s − 0.232·22-s − 0.00850·24-s + 0.00800·25-s + 0.331·26-s − 0.00712·27-s + 0.691·29-s − 0.00608·30-s + 0.0926·31-s + 0.00552·32-s + 0.126·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{3} \cdot 3^{3} \cdot 5^{3} \cdot 17^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+3^{ -s})^{-1}(1-5^{- s})^{-1}(1-17^{- s})^{-1}\prod_{p \nmid 510 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.