Properties

Degree 4
Conductor $ 2^{3} \cdot 5^{3} \cdot 7^{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  + 0.353·2-s − 1.73·3-s + 0.125·4-s + 0.0894·5-s − 0.612·6-s + 0.0441·8-s + 9-s + 0.0316·10-s + 0.986·11-s − 0.216·12-s − 0.154·15-s + 0.0156·16-s − 1.02·17-s + 0.353·18-s − 0.144·19-s + 0.0111·20-s + 0.348·22-s − 1.00·23-s − 0.0765·24-s + 0.00800·25-s + 1.32·29-s − 0.0547·30-s − 1.06·31-s + 0.00552·32-s − 1.70·33-s − 0.363·34-s + 0.125·36-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2401000 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & -\, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2401000\)    =    \(2^{3} \cdot 5^{3} \cdot 7^{4}\)
\( \varepsilon \)  =  $-1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2401000,\ (\ :1.5, 0.5),\ -1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1-5^{- s})^{-1} \prod_{p \nmid 490 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.