Properties

Degree 3
Conductor $ 2^{4} \cdot 7^{2} \cdot 19^{2} \cdot 47^{2} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

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Normalization:  

(not yet available)

Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 0.666·3-s − 0.800·5-s + 0.142·7-s + 1.11·9-s + 0.454·11-s + 0.230·13-s + 0.533·15-s − 0.0588·17-s + 0.0526·19-s − 0.0952·21-s − 0.304·23-s + 1.43·25-s − 0.185·27-s − 0.689·29-s − 0.870·31-s − 0.303·33-s − 0.114·35-s + 0.729·37-s − 0.153·39-s − 0.902·41-s + 0.488·43-s − 0.888·45-s + 0.0212·47-s + 0.0204·49-s + 0.0392·51-s + 0.207·53-s − 0.363·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s,E,\mathrm{sym}^{2})=\mathstrut &\left(2^{4} \cdot 7^{2} \cdot 19^{2} \cdot 47^{2}\right)^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr=\mathstrut & \,\Lambda(1-{s}, E,\mathrm{sym}^{2})\end{aligned}\]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(2^{4} \cdot 7^{2} \cdot 19^{2} \cdot 47^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  \((3,\ 2^{4} \cdot 7^{2} \cdot 19^{2} \cdot 47^{2} ,\ ( 1 : 1.0 ),\ 1 )\)

Euler product

\[\begin{aligned}L(s, E, \mathrm{sym}^{2}) = (1-7^{- s})^{-1}(1-19^{- s})^{-1}(1-47^{- s})^{-1}\prod_{p \nmid 100016 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1}\end{aligned}\]

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.