Properties

Degree 4
Conductor $ 197^{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  + 0.809·7-s − 2·9-s − 0.657·11-s + 0.938·13-s − 16-s − 3.42·17-s + 1.05·19-s + 1.00·23-s − 2·25-s − 0.403·29-s − 2.20·31-s − 0.777·37-s − 0.0342·41-s − 0.301·43-s − 0.921·47-s + 0.860·49-s − 0.155·53-s − 1.01·61-s − 1.61·63-s + 0.619·67-s − 1.04·71-s − 1.05·73-s − 0.532·77-s − 0.438·79-s + 3·81-s + 1.08·83-s + 1.08·89-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7645373 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(7645373\)    =    \(197^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 7645373,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+197^{ -s})^{-1}\prod_{p \nmid 197 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.