Properties

Degree 4
Conductor $ 37^{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  − 1.73·3-s + 1.07·5-s + 0.701·7-s + 9-s − 0.411·11-s + 0.938·13-s − 1.85·15-s − 16-s − 1.21·21-s − 0.761·23-s + 0.912·25-s − 0.845·29-s + 1.06·31-s + 0.712·33-s + 0.753·35-s − 0.00444·37-s − 1.62·39-s + 0.0342·41-s − 0.581·43-s + 1.07·45-s + 0.363·47-s + 1.73·48-s − 1.09·49-s − 0.272·53-s − 0.441·55-s − 0.953·59-s + 0.973·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 50653 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(50653\)    =    \(37^{3}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 50653,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+37^{ -s})^{-1}\prod_{p \nmid 37 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.