Properties

Degree 4
Conductor $ 2^{12} \cdot 431^{3} $
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  − 1.73·3-s + 0.268·5-s − 0.431·7-s + 9-s − 0.411·11-s + 0.853·13-s − 0.464·15-s − 0.171·17-s − 0.929·19-s + 0.748·21-s − 1.00·23-s + 0.232·25-s + 0.941·29-s + 1.06·31-s + 0.712·33-s − 0.115·35-s + 0.355·37-s − 1.47·39-s + 1.05·41-s + 0.624·43-s + 0.268·45-s + 1.08·47-s − 0.180·49-s + 0.296·51-s + 0.272·53-s − 0.110·55-s + 1.61·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{12} \cdot 431^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{12} \cdot 431^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{12} \cdot 431^{3} ,\ ( \ : 1.5, 0.5 ),\ 1 )$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-431^{- s})^{-1}\prod_{p \nmid 27584 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.