# Properties

 Degree 2 Conductor $2 \cdot 5$ Sign $1$ Primitive yes Self-dual yes

# Related objects

## Dirichlet series

 $L(s,f)$  = 1 − 0.707·2-s − 0.894·3-s + 0.5·4-s + 0.447·5-s + 0.632·6-s + 0.999·7-s − 0.353·8-s − 0.199·9-s − 0.316·10-s + 1.16·11-s − 0.447·12-s + 1.77·13-s − 0.706·14-s − 0.400·15-s + 0.250·16-s + 1.52·17-s + 0.140·18-s + 1.67·19-s + 0.223·20-s − 0.894·21-s − 0.820·22-s − 0.0700·23-s + 0.316·24-s + 0.199·25-s − 1.25·26-s + 1.07·27-s + 0.499·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s,f)=\mathstrut & 10 ^{s/2} \, \Gamma_{\R}(s+(1 + 18.6i)) \, \Gamma_{\R}(s+(1 - 18.6i)) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$10$$    =    $$2 \cdot 5$$ $$\varepsilon$$ = $1$ primitive : yes self-dual : yes Selberg data = $$(2,\ 10,\ (1 + 18.6257215488i, 1 - 18.6257215488i:\ ),\ 1)$$

## Euler product

\begin{aligned}L(s,f) = \prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\end{aligned}