Properties

Degree $4$
Conductor $194481000$
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  − 0.353·2-s + 0.125·4-s − 0.0894·5-s − 0.0441·8-s + 0.0316·10-s − 0.657·11-s − 0.938·13-s + 0.0156·16-s − 3.42·17-s + 0.144·19-s − 0.0111·20-s + 0.232·22-s − 1.08·23-s + 0.00800·25-s + 0.331·26-s − 0.845·29-s − 1.06·31-s − 0.00552·32-s + 1.21·34-s − 1.15·37-s − 0.0512·38-s + 0.00395·40-s + 1.00·41-s − 0.993·43-s − 0.0822·44-s + 0.384·46-s + 0.968·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{3} \cdot 3^{4} \cdot 5^{3} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2^{3} \cdot 3^{4} \cdot 5^{3} \cdot 7^{4}\)
Sign: $-1$
Primitive: yes
Self-dual: yes
Selberg data: \((4,\ 2^{3} \cdot 3^{4} \cdot 5^{3} \cdot 7^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )\)

Particular Values

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

Euler product

\(L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1} (1+5^{ -s})^{-1} \prod_{p \nmid 4410 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\)

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.