# Properties

 Degree 3 Conductor $197^{2}$ Sign $1$ Motivic weight 2 Primitive yes Self-dual yes

# Related objects

(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.