Properties

Degree 4
Conductor $ 2^{2} \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  − 0.769·3-s − 4-s + 1.07·7-s + 0.814·9-s + 0.769·12-s + 0.938·13-s + 16-s + 0.171·17-s + 1.06·19-s − 0.831·21-s + 0.543·23-s − 1.56·27-s − 1.07·28-s − 0.845·29-s + 1.06·31-s − 0.814·36-s + 0.622·37-s − 0.722·39-s − 1.05·41-s + 0.496·43-s − 1.08·47-s − 0.769·48-s + 0.553·49-s − 0.131·51-s − 0.938·52-s − 1.08·53-s − 0.817·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1.5) \, \Gamma_{\C}(s+0.5) \, L(s, E, \mathrm{sym}^{3})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{3}) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(2500\)    =    \(2^{2} \cdot 5^{4}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(4,\ 2500,\ (\ :1.5, 0.5),\ 1)$

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{3}) = (1+8\ 2^{-2 s})^{-1} \prod_{p \nmid 100 }\prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}} \right)^{-1}\end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{3}) \approx 0.8459184997\] \[L(1, E, \mathrm{sym}^{3}) \approx 0.8236930003\]

Imaginary part of the first few zeros on the critical line

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