Properties

Degree $4$
Conductor $874302741$
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  − 1.06·2-s + 0.375·4-s − 1.07·5-s − 0.662·8-s + 1.13·10-s + 0.657·11-s + 0.0213·13-s + 0.546·16-s − 0.0142·17-s + 1.06·19-s − 0.402·20-s − 0.697·22-s + 0.912·25-s − 0.0226·26-s − 0.691·29-s − 0.0926·31-s + 0.580·32-s + 0.0151·34-s + 0.622·37-s − 1.12·38-s + 0.711·40-s + 0.594·41-s + 0.993·43-s + 0.246·44-s + 0.744·47-s − 2·49-s − 0.967·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{4} \cdot 13^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3^{4} \cdot 13^{3} \cdot 17^{3}\)
Sign: $1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 3^{4} \cdot 13^{3} \cdot 17^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-13^{- s})^{-1}(1+17^{ -s})^{-1}\prod_{p \nmid 1989 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.