Properties

Degree $4$
Conductor $1.002\times 10^{12}$
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  − 1.06·2-s − 0.192·3-s + 0.375·4-s + 0.0894·5-s + 0.204·6-s − 0.662·8-s + 0.0370·9-s − 0.0948·10-s − 0.657·11-s − 0.0721·12-s + 0.938·13-s − 0.0172·15-s + 0.546·16-s − 0.171·17-s − 0.0392·18-s − 1.06·19-s + 0.0335·20-s + 0.697·22-s − 0.00906·23-s + 0.127·24-s + 0.00800·25-s − 0.995·26-s − 0.00712·27-s + 0.00640·29-s + 0.0182·30-s − 1.06·31-s + 0.580·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 3^{3} \cdot 5^{3} \cdot 23^{3} \cdot 29^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+3^{ -s})^{-1}(1-5^{- s})^{-1}(1+23^{ -s})^{-1}(1-29^{- s})^{-1}\prod_{p \nmid 10005 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.