Properties

Degree 4
Conductor $ 2^{10} \cdot 5^{4} \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 + 0.431·7-s + 9-s + 0.411·11-s + 0.853·13-s + 0.171·17-s + 0.929·19-s − 0.748·21-s + 1.00·23-s − 0.941·29-s + 1.06·31-s − 0.712·33-s + 0.355·37-s − 1.47·39-s + 1.05·41-s + 0.624·43-s − 1.08·47-s − 0.180·49-s − 0.296·51-s + 0.272·53-s − 1.61·57-s − 0.734·59-s − 0.495·61-s + 0.431·63-s − 0.619·67-s − 1.74·69-s + 1.04·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{10} \cdot 5^{4} \cdot 431^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{10} \cdot 5^{4} \cdot 431^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{10} \cdot 5^{4} \cdot 431^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1-431^{- s})^{-1}\prod_{p \nmid 344800 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.