# Properties

 Label 2-819-91.20-c1-0-21 Degree $2$ Conductor $819$ Sign $0.755 - 0.655i$ Analytic cond. $6.53974$ Root an. cond. $2.55729$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.11 − 0.299i)2-s + (−0.576 + 0.332i)4-s + (−0.549 + 0.549i)5-s + (1.61 − 2.09i)7-s + (−2.17 + 2.17i)8-s + (−0.448 + 0.777i)10-s + (0.824 + 3.07i)11-s + (2.63 + 2.45i)13-s + (1.17 − 2.82i)14-s + (−1.11 + 1.92i)16-s + (−1.74 − 3.03i)17-s + (6.06 + 1.62i)19-s + (0.133 − 0.499i)20-s + (1.83 + 3.18i)22-s + (4.89 + 2.82i)23-s + ⋯
 L(s)  = 1 + (0.789 − 0.211i)2-s + (−0.288 + 0.166i)4-s + (−0.245 + 0.245i)5-s + (0.609 − 0.792i)7-s + (−0.769 + 0.769i)8-s + (−0.141 + 0.245i)10-s + (0.248 + 0.927i)11-s + (0.731 + 0.682i)13-s + (0.313 − 0.754i)14-s + (−0.278 + 0.482i)16-s + (−0.424 − 0.735i)17-s + (1.39 + 0.372i)19-s + (0.0299 − 0.111i)20-s + (0.392 + 0.679i)22-s + (1.01 + 0.588i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$819$$    =    $$3^{2} \cdot 7 \cdot 13$$ Sign: $0.755 - 0.655i$ Analytic conductor: $$6.53974$$ Root analytic conductor: $$2.55729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{819} (748, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 819,\ (\ :1/2),\ 0.755 - 0.655i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.90848 + 0.712316i$$ $$L(\frac12)$$ $$\approx$$ $$1.90848 + 0.712316i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (-1.61 + 2.09i)T$$
13 $$1 + (-2.63 - 2.45i)T$$
good2 $$1 + (-1.11 + 0.299i)T + (1.73 - i)T^{2}$$
5 $$1 + (0.549 - 0.549i)T - 5iT^{2}$$
11 $$1 + (-0.824 - 3.07i)T + (-9.52 + 5.5i)T^{2}$$
17 $$1 + (1.74 + 3.03i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-6.06 - 1.62i)T + (16.4 + 9.5i)T^{2}$$
23 $$1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 + (4.54 - 7.87i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (0.888 - 0.888i)T - 31iT^{2}$$
37 $$1 + (-0.151 - 0.564i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (0.704 + 2.63i)T + (-35.5 + 20.5i)T^{2}$$
43 $$1 + (-6.60 + 3.81i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (0.267 + 0.267i)T + 47iT^{2}$$
53 $$1 - 11.6T + 53T^{2}$$
59 $$1 + (0.635 - 2.37i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (6.70 - 3.86i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (6.90 - 1.85i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 + (-2.51 + 9.37i)T + (-61.4 - 35.5i)T^{2}$$
73 $$1 + (7.71 + 7.71i)T + 73iT^{2}$$
79 $$1 + 10.8T + 79T^{2}$$
83 $$1 + (10.3 - 10.3i)T - 83iT^{2}$$
89 $$1 + (-7.02 + 1.88i)T + (77.0 - 44.5i)T^{2}$$
97 $$1 + (0.704 + 0.188i)T + (84.0 + 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$