Properties

Degree $4$
Conductor $21952$
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  − 0.769·3-s − 2.14·5-s + 0.0539·7-s + 0.814·9-s + 1.65·15-s + 0.856·17-s + 0.821·19-s − 0.0415·21-s + 1.30·23-s + 1.96·25-s − 1.56·27-s − 0.691·29-s − 1.06·31-s − 0.115·35-s + 1.01·37-s + 0.594·41-s − 0.624·43-s − 1.74·45-s + 0.968·47-s + 0.00291·49-s − 0.658·51-s + 0.155·53-s − 0.632·57-s − 1.08·59-s − 0.889·61-s + 0.0439·63-s − 0.218·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 21952 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21952\)    =    \(2^{6} \cdot 7^{3}\)
Sign: $-1$
Arithmetic: yes
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 21952,\ (\ :1.5, 0.5),\ -1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-7^{- s})^{-1}\prod_{p \nmid 56 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.