Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{4} \cdot 41^{3} $
Sign $-1$
Motivic weight 3
Primitive yes
Self-dual yes

Related objects

Learn more about

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  − 0.962·3-s − 4-s − 0.804·5-s − 0.185·9-s − 0.411·11-s + 0.962·12-s − 0.938·13-s + 0.774·15-s + 16-s − 1.07·17-s − 1.05·19-s + 0.804·20-s + 0.407·23-s − 0.792·25-s + 0.285·27-s − 2.68·29-s + 3.76·31-s + 0.395·33-s + 0.185·36-s − 0.777·37-s + 0.903·39-s + 0.00380·41-s − 0.624·43-s + 0.411·44-s + 0.149·45-s − 0.977·47-s − 0.962·48-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(2^{2} \cdot 7^{4} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2^{2} \cdot 7^{4} \cdot 41^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2^{2} \cdot 7^{4} \cdot 41^{3} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1} (1-41^{- s})^{-1}\prod_{p \nmid 8036 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.