Properties

Degree 3
Conductor $ 197^{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  + 2-s − 3-s − 5-s − 6-s + 0.285·7-s + 2·9-s − 10-s + 0.454·11-s − 0.692·13-s + 0.285·14-s + 15-s + 16-s + 2.76·17-s + 2·18-s − 0.526·19-s − 0.285·21-s + 0.454·22-s − 0.608·23-s + 2·25-s − 0.692·26-s − 2·27-s + 0.689·29-s + 30-s + 2.22·31-s + 32-s − 0.454·33-s + 2.76·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s,E,\mathrm{sym}^{2})=\mathstrut & 38809 ^{s/2} \, \Gamma_{\R}(s+1) \, \Gamma_{\C}(s+1) \, L(s, E, \mathrm{sym}^{2})\cr =\mathstrut & \, \Lambda(1-{s}, E,\mathrm{sym}^{2}) \end{aligned} \]

Invariants

\( d \)  =  \(3\)
\( N \)  =  \(38809\)    =    \(197^{2}\)
\( \varepsilon \)  =  $1$
primitive  :  yes
self-dual  :  yes
Selberg data  =  $(3,\ 38809,\ (1:1.0),\ 1)$

Euler product

\[\begin{aligned} L(s, E, \mathrm{sym}^{2}) = (1-197^{- s})^{-1}\prod_{p \nmid 197 }\prod_{j=0}^{2} \left(1- \frac{\alpha_p^j\beta_p^{2-j}}{p^{s}} \right)^{-1} \end{aligned}\]

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.