Properties

Degree 4
Conductor $ 2^{3} \cdot 13^{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 − 1.73·3-s + 0.125·4-s + 0.804·5-s − 0.612·6-s − 0.701·7-s + 0.0441·8-s + 9-s + 0.284·10-s + 0.986·11-s − 0.216·12-s − 0.0213·13-s − 0.248·14-s − 1.39·15-s + 0.0156·16-s + 1.07·17-s + 0.353·18-s − 0.144·19-s + 0.100·20-s + 1.21·21-s + 0.348·22-s + 1.08·23-s − 0.0765·24-s − 0.792·25-s − 0.00754·26-s − 0.0877·28-s − 0.691·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 17576 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(17576\)    =    \(2^{3} \cdot 13^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 17576,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1-2^{- s})^{-1}(1+13^{ -s})^{-1}\prod_{p \nmid 26 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.