Properties

Degree 4
Conductor $ 2^{12} \cdot 3^{3} \cdot 7^{4} $
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 + 1.07·5-s + 0.0370·9-s + 0.657·11-s + 0.938·13-s + 0.206·15-s + 0.171·17-s − 1.06·19-s + 0.912·25-s + 0.00712·27-s − 0.691·29-s + 0.126·33-s + 1.01·37-s + 0.180·39-s + 0.594·41-s − 0.993·43-s + 0.0397·45-s + 0.0329·51-s + 1.08·53-s + 0.706·55-s − 0.204·57-s + 0.688·59-s + 0.495·61-s + 1.00·65-s + 0.860·67-s − 1.05·73-s + 0.175·75-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{12} \cdot 3^{3} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{12} \cdot 3^{3} \cdot 7^{4} ,\ ( \ : 1.5, 0.5 ),\ 1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1} \prod_{p \nmid 9408 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

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

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.