# Properties

 Label 2-819-91.4-c1-0-44 Degree $2$ Conductor $819$ Sign $0.330 - 0.943i$ 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 − 2.30i·2-s − 3.30·4-s + (−0.733 + 0.423i)5-s + (−0.357 − 2.62i)7-s + 3.00i·8-s + (0.975 + 1.69i)10-s + (−1.30 + 0.751i)11-s + (−2.92 + 2.11i)13-s + (−6.03 + 0.824i)14-s + 0.313·16-s + 2.07·17-s + (0.0410 + 0.0237i)19-s + (2.42 − 1.40i)20-s + (1.73 + 2.99i)22-s − 7.81·23-s + ⋯
 L(s)  = 1 − 1.62i·2-s − 1.65·4-s + (−0.328 + 0.189i)5-s + (−0.135 − 0.990i)7-s + 1.06i·8-s + (0.308 + 0.534i)10-s + (−0.392 + 0.226i)11-s + (−0.810 + 0.585i)13-s + (−1.61 + 0.220i)14-s + 0.0782·16-s + 0.502·17-s + (0.00942 + 0.00544i)19-s + (0.542 − 0.313i)20-s + (0.369 + 0.639i)22-s − 1.63·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$819$$    =    $$3^{2} \cdot 7 \cdot 13$$ Sign: $0.330 - 0.943i$ Analytic conductor: $$6.53974$$ Root analytic conductor: $$2.55729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{819} (550, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 819,\ (\ :1/2),\ 0.330 - 0.943i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.158570 + 0.112489i$$ $$L(\frac12)$$ $$\approx$$ $$0.158570 + 0.112489i$$ $$L(\frac{3}{2})$$ 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 + (0.357 + 2.62i)T$$
13 $$1 + (2.92 - 2.11i)T$$
good2 $$1 + 2.30iT - 2T^{2}$$
5 $$1 + (0.733 - 0.423i)T + (2.5 - 4.33i)T^{2}$$
11 $$1 + (1.30 - 0.751i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 - 2.07T + 17T^{2}$$
19 $$1 + (-0.0410 - 0.0237i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + 7.81T + 23T^{2}$$
29 $$1 + (-0.679 + 1.17i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (6.80 + 3.93i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + 6.70iT - 37T^{2}$$
41 $$1 + (-8.67 - 5.00i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (-0.311 + 0.180i)T + (23.5 - 40.7i)T^{2}$$
53 $$1 + (-1.35 + 2.34i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 - 1.64iT - 59T^{2}$$
61 $$1 + (2.26 - 3.91i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.76 - 1.02i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (12.3 - 7.10i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-5.85 - 3.38i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (5.82 + 10.0i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 11.5iT - 83T^{2}$$
89 $$1 + 17.5iT - 89T^{2}$$
97 $$1 + (-0.369 + 0.213i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$