Properties

Degree $4$
Conductor $237276$
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  + 0.192·3-s − 4-s − 1.07·7-s + 0.0370·9-s − 0.192·12-s + 0.0213·13-s + 16-s − 0.171·17-s − 0.821·19-s − 0.207·21-s − 2·25-s + 0.00712·27-s + 1.07·28-s + 0.845·29-s − 0.672·31-s − 0.0370·36-s − 0.622·37-s + 0.00410·39-s − 2.83·41-s + 0.993·43-s + 0.192·48-s + 0.553·49-s − 0.0329·51-s − 0.0213·52-s − 1.08·53-s − 0.158·57-s + 0.688·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 237276 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(237276\)    =    \(2^{2} \cdot 3^{3} \cdot 13^{3}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 237276,\ (\ :1.5, 0.5),\ 1)\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1}(1-3^{- s})^{-1}(1-13^{- s})^{-1}\prod_{p \nmid 156 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.