Properties

Degree 4
Conductor $ 2^{2} \cdot 3^{4} \cdot 5^{4} \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  − 4-s − 0.938·13-s + 16-s + 0.171·17-s − 1.06·19-s − 0.543·23-s + 0.845·29-s − 1.06·31-s + 0.622·37-s − 1.05·41-s + 0.496·43-s − 1.08·47-s + 0.938·52-s + 1.08·53-s + 0.688·59-s + 0.495·61-s − 64-s + 0.474·67-s − 0.171·68-s + 0.0401·71-s − 0.455·73-s + 1.06·76-s − 1.07·79-s + 1.03·83-s + 1.01·89-s + 0.543·92-s − 0.397·97-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{2} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} ,\ ( \ : 1.5, 0.5 ),\ 1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1} \prod_{p \nmid 44100 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.