Properties

Degree 4
Conductor $ 2^{6} \cdot 19^{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  + 0.769·3-s + 0.804·5-s + 0.809·7-s + 0.814·9-s + 1.06·11-s + 0.853·13-s + 0.619·15-s − 0.642·17-s − 0.0120·19-s + 0.623·21-s − 0.792·25-s + 1.56·27-s − 0.691·29-s + 0.0926·31-s + 0.822·33-s + 0.651·35-s − 1.15·37-s + 0.656·39-s − 1.05·41-s + 0.918·43-s + 0.655·45-s + 0.363·47-s + 0.860·49-s − 0.494·51-s + 0.870·53-s + 0.860·55-s − 0.00929·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 438976 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(438976\)    =    \(2^{6} \cdot 19^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 438976,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+19^{ -s})^{-1}\prod_{p \nmid 152 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.