Properties

Degree 4
Conductor $ 3^{3} \cdot 5^{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  + 0.192·3-s − 0.809·7-s + 0.0370·9-s − 0.986·11-s + 0.533·13-s − 16-s + 0.856·17-s + 0.784·19-s − 0.155·21-s + 0.543·23-s + 0.00712·27-s + 2.68·29-s + 0.921·31-s − 0.189·33-s + 0.622·37-s + 0.102·39-s + 0.548·41-s + 0.301·43-s + 0.558·47-s − 0.192·48-s + 0.860·49-s + 0.164·51-s − 0.933·53-s + 0.151·57-s + 0.397·59-s − 1.07·61-s − 0.0299·63-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 16875 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(16875\)    =    \(3^{3} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 16875,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-3^{- s})^{-1} \prod_{p \nmid 75 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.