Properties

Degree $4$
Conductor $78953589$
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 + 1.07·5-s + 0.204·6-s + 0.662·8-s + 0.0370·9-s + 1.13·10-s − 0.0274·11-s + 0.0721·12-s + 0.0213·13-s + 0.206·15-s + 0.546·16-s − 0.171·17-s + 0.0392·18-s + 1.06·19-s + 0.402·20-s − 0.0290·22-s − 1.30·23-s + 0.127·24-s + 0.912·25-s + 0.0226·26-s + 0.00712·27-s − 2.68·29-s + 0.219·30-s − 0.580·32-s − 0.00527·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 78953589 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78953589\)    =    \(3^{3} \cdot 11^{3} \cdot 13^{3}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 78953589,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1}(1+11^{ -s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 429 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.