Properties

Degree 3
Conductor $ 2^{2} \cdot 19^{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  + 0.5·2-s − 0.666·3-s + 0.250·4-s + 2.20·5-s − 0.333·6-s + 0.285·7-s + 0.125·8-s + 1.11·9-s + 1.10·10-s − 0.636·11-s − 0.166·12-s − 0.923·13-s + 0.142·14-s − 1.46·15-s + 0.0625·16-s − 0.470·17-s + 0.555·18-s + 0.0526·19-s + 0.550·20-s − 0.190·21-s − 0.318·22-s − 0.956·23-s − 0.0833·24-s + 2.64·25-s − 0.461·26-s − 0.185·27-s + 0.0714·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 1444 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(1444\)    =    \(2^{2} \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 1444,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-2^{- s})^{-1}(1-19^{- s})^{-1}\prod_{p \nmid 38 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 2.191841609\] \[L(1, E, \mathrm{sym}^{2}) \approx 1.595690906\]

Imaginary part of the first few zeros on the critical line

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