# Properties

 Label 2-800-5.4-c3-0-5 Degree $2$ Conductor $800$ Sign $-0.894 + 0.447i$ Analytic cond. $47.2015$ Root an. cond. $6.87033$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.47i·3-s + 31.3i·7-s + 6.99·9-s − 8.94·11-s − 62i·13-s + 46i·17-s − 107.·19-s − 140·21-s + 192. i·23-s + 152. i·27-s + 90·29-s − 152.·31-s − 40.0i·33-s + 214i·37-s + 277.·39-s + ⋯
 L(s)  = 1 + 0.860i·3-s + 1.69i·7-s + 0.259·9-s − 0.245·11-s − 1.32i·13-s + 0.656i·17-s − 1.29·19-s − 1.45·21-s + 1.74i·23-s + 1.08i·27-s + 0.576·29-s − 0.880·31-s − 0.211i·33-s + 0.950i·37-s + 1.13·39-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$800$$    =    $$2^{5} \cdot 5^{2}$$ Sign: $-0.894 + 0.447i$ Analytic conductor: $$47.2015$$ Root analytic conductor: $$6.87033$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{800} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 800,\ (\ :3/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.9737835207$$ $$L(\frac12)$$ $$\approx$$ $$0.9737835207$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1$$
good3 $$1 - 4.47iT - 27T^{2}$$
7 $$1 - 31.3iT - 343T^{2}$$
11 $$1 + 8.94T + 1.33e3T^{2}$$
13 $$1 + 62iT - 2.19e3T^{2}$$
17 $$1 - 46iT - 4.91e3T^{2}$$
19 $$1 + 107.T + 6.85e3T^{2}$$
23 $$1 - 192. iT - 1.21e4T^{2}$$
29 $$1 - 90T + 2.43e4T^{2}$$
31 $$1 + 152.T + 2.97e4T^{2}$$
37 $$1 - 214iT - 5.06e4T^{2}$$
41 $$1 + 10T + 6.89e4T^{2}$$
43 $$1 + 67.0iT - 7.95e4T^{2}$$
47 $$1 + 398. iT - 1.03e5T^{2}$$
53 $$1 + 678iT - 1.48e5T^{2}$$
59 $$1 - 411.T + 2.05e5T^{2}$$
61 $$1 - 250T + 2.26e5T^{2}$$
67 $$1 + 49.1iT - 3.00e5T^{2}$$
71 $$1 + 366.T + 3.57e5T^{2}$$
73 $$1 - 522iT - 3.89e5T^{2}$$
79 $$1 + 876.T + 4.93e5T^{2}$$
83 $$1 - 380. iT - 5.71e5T^{2}$$
89 $$1 + 970T + 7.04e5T^{2}$$
97 $$1 - 934iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$