Properties

Degree $4$
Conductor $12446784$
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  + 1.07·5-s + 2.30·11-s + 1.28·13-s − 0.856·17-s + 1.06·19-s − 0.761·23-s + 0.912·25-s + 0.307·29-s + 1.06·31-s + 1.01·37-s − 0.685·41-s + 0.993·43-s − 0.968·47-s + 0.933·53-s + 2.47·55-s + 0.688·59-s − 0.495·61-s + 1.37·65-s + 0.218·67-s − 1.06·71-s − 0.455·73-s + 1.07·79-s − 0.918·85-s + 0.300·89-s + 1.14·95-s − 0.397·97-s + 2.16·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12446784\)    =    \(2^{6} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 12446784,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = \prod_{p \nmid 3528 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.