Properties

Degree 4
Conductor $ 2^{3} \cdot 3^{4} \cdot 5^{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  − 0.353·2-s + 0.125·4-s + 0.0894·5-s − 1.07·7-s − 0.0441·8-s − 0.0316·10-s + 2.30·11-s + 0.853·13-s + 0.381·14-s + 0.0156·16-s − 0.171·17-s + 1.06·19-s + 0.0111·20-s − 0.814·22-s + 0.00800·25-s − 0.301·26-s − 0.134·28-s + 0.845·29-s + 1.06·31-s − 0.00552·32-s + 0.0605·34-s − 0.0965·35-s − 0.355·37-s − 0.375·38-s − 0.00395·40-s − 0.624·43-s + 0.287·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 81000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(81000\)    =    \(2^{3} \cdot 3^{4} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 81000,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+2^{ -s})^{-1} (1-5^{- s})^{-1}\prod_{p \nmid 90 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.