Properties

Degree 4
Conductor $ 53^{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  + 1.06·2-s − 1.73·3-s + 0.375·4-s − 1.83·6-s − 0.431·7-s + 0.662·8-s + 9-s − 0.649·12-s + 1.08·13-s − 0.458·14-s + 0.546·16-s + 1.07·17-s + 1.06·18-s + 0.784·19-s + 0.748·21-s + 0.190·23-s − 1.14·24-s − 2·25-s + 1.15·26-s − 0.161·28-s + 0.403·29-s − 1.06·31-s − 0.580·32-s + 1.13·34-s + 0.375·36-s − 1.08·37-s + 0.832·38-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 148877 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(148877\)    =    \(53^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 148877,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+53^{ -s})^{-1}\prod_{p \nmid 53 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.