Properties

Degree 4
Conductor 50653
Sign $1$
Self-dual yes
Motivic weight 3

Related objects

Downloads

Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{3})$  = 1  − 0.962·3-s − 2·4-s + 0.701·7-s − 0.185·9-s − 1.068·11-s + 1.924·12-s + 0.853·13-s + 3·16-s + 0.171·17-s − 0.821·19-s − 0.675·21-s − 0.543·23-s − 2·25-s + 0.285·27-s − 1.403·28-s + 0.845·29-s + 1.066·31-s + 1.028·33-s + 0.370·36-s + 0.004·37-s − 0.821·39-s + 0.034·41-s − 0.624·43-s + 2.137·44-s − 0.791·47-s − 2.886·48-s − 1.099·49-s + ⋯

Functional equation

\[\begin{align} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 50653 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{align} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(50653\)    =    \(37^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 50653,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{equation} L(s, E, \mathrm{sym}^{3}) = (1-37^{- s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{equation}\]

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.