Properties

Degree 4
Conductor $ 3^{4} \cdot 11^{3} \cdot 19^{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  − 2·4-s − 1.07·7-s − 0.0274·11-s − 0.533·13-s + 3·16-s + 1.07·17-s + 0.543·23-s − 2·25-s + 2.15·28-s − 0.0926·31-s + 0.622·37-s − 1.05·41-s − 0.624·43-s + 0.0548·44-s − 1.08·47-s + 0.553·49-s + 1.06·52-s − 0.583·53-s − 0.721·59-s + 0.461·61-s − 4·64-s − 0.619·67-s − 2.14·68-s + 0.666·71-s + 0.833·73-s + 0.0296·77-s + 0.203·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut &\left(3^{4} \cdot 11^{3} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr=\mathstrut & -\,\Lambda(1-{s}, E,\mathrm{sym}^{3})\end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(3^{4} \cdot 11^{3} \cdot 19^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 3^{4} \cdot 11^{3} \cdot 19^{4} ,\ ( \ : 1.5, 0.5 ),\ -1 )$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+11^{ -s})^{-1} \prod_{p \nmid 35739 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.