Properties

Degree $4$
Conductor $48316878848$
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  + 0.769·3-s + 0.814·9-s − 2.30·11-s − 0.938·13-s + 0.0142·17-s + 1.06·19-s − 2·25-s + 1.56·27-s − 1.06·31-s − 1.77·33-s − 1.03·37-s − 0.722·39-s + 1.05·41-s + 0.624·43-s + 0.0109·51-s − 1.08·53-s + 0.817·57-s + 0.889·61-s + 1.02·67-s + 0.455·73-s − 1.53·75-s − 1.07·79-s + 0.980·81-s − 1.01·89-s − 0.820·93-s − 0.0293·97-s − 1.87·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 7^{4} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{12} \cdot 7^{4} \cdot 17^{3}\)
Sign: $1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{12} \cdot 7^{4} \cdot 17^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1-17^{- s})^{-1}\prod_{p \nmid 53312 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.