Properties

Degree 4
Conductor $ 19^{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  + 0.769·3-s − 2·4-s − 0.268·5-s + 0.701·7-s + 0.814·9-s − 1.06·11-s − 1.53·12-s + 0.853·13-s − 0.206·15-s + 3·16-s + 1.07·17-s + 0.0120·19-s + 0.536·20-s + 0.540·21-s + 0.232·25-s + 1.56·27-s − 1.40·28-s − 0.845·29-s + 1.06·31-s − 0.822·33-s − 0.188·35-s − 1.62·36-s − 0.622·37-s + 0.656·39-s + 1.05·41-s + 0.301·43-s + 2.13·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 6859 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(6859\)    =    \(19^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 6859,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-19^{- s})^{-1}\prod_{p \nmid 19 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 1.201026362\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.9919550069\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line