Properties

Degree 3
Conductor $ 19^{2} $
Sign $1$
Motivic weight 2
Primitive yes
Self-dual yes

Related objects

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Normalization:  

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Dirichlet series

$L(s, E, \mathrm{sym}^{2})$  = 1  − 2-s + 0.333·3-s + 2·4-s + 0.800·5-s − 0.333·6-s − 0.857·7-s − 2·8-s − 0.222·9-s − 0.800·10-s − 0.181·11-s + 0.666·12-s + 0.230·13-s + 0.857·14-s + 0.266·15-s + 3·16-s − 0.470·17-s + 0.222·18-s + 0.0526·19-s + 1.60·20-s − 0.285·21-s + 0.181·22-s − 23-s − 0.666·24-s − 0.160·25-s − 0.230·26-s + 0.814·27-s − 1.71·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(361\)    =    \(19^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 361,\ (1:1.0),\ 1)$

Euler product

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

Particular Values

\[L(1/2, E, \mathrm{sym}^{2}) \approx 0.9630442737\] \[L(1, E, \mathrm{sym}^{2}) \approx 0.9279031293\]

Imaginary part of the first few zeros on the critical line

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