Properties

Degree 4
Conductor $ 2^{2} \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  + 0.769·3-s − 4-s − 0.0894·5-s − 1.07·7-s + 0.814·9-s − 0.769·12-s − 0.938·13-s − 0.0688·15-s + 16-s − 0.171·17-s + 1.06·19-s + 0.0894·20-s − 0.831·21-s − 0.543·23-s + 0.00800·25-s + 1.56·27-s + 1.07·28-s − 0.845·29-s + 1.06·31-s + 0.0965·35-s − 0.814·36-s − 0.622·37-s − 0.722·39-s − 1.05·41-s − 0.496·43-s − 0.0728·45-s + 1.08·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(500\)    =    \(2^{2} \cdot 5^{3}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 500,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1}(1+5^{ -s})^{-1}\prod_{p \nmid 20 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0.7049320831\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.8902757968\]

Imaginary part of the first few zeros on the critical line

Graph of the $Z$-function along the critical line