# Properties

 Label 2-1800-5.4-c3-0-28 Degree $2$ Conductor $1800$ Sign $0.894 + 0.447i$ Analytic cond. $106.203$ Root an. cond. $10.3055$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·7-s − 72·11-s + 6i·13-s − 38i·17-s − 52·19-s + 152i·23-s − 78·29-s + 120·31-s − 150i·37-s − 362·41-s + 484i·43-s − 280i·47-s + 327·49-s − 670i·53-s + 696·59-s + ⋯
 L(s)  = 1 + 0.215i·7-s − 1.97·11-s + 0.128i·13-s − 0.542i·17-s − 0.627·19-s + 1.37i·23-s − 0.499·29-s + 0.695·31-s − 0.666i·37-s − 1.37·41-s + 1.71i·43-s − 0.868i·47-s + 0.953·49-s − 1.73i·53-s + 1.53·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{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: $$1800$$    =    $$2^{3} \cdot 3^{2} \cdot 5^{2}$$ Sign: $0.894 + 0.447i$ Analytic conductor: $$106.203$$ Root analytic conductor: $$10.3055$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{1800} (649, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1800,\ (\ :3/2),\ 0.894 + 0.447i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.226501376$$ $$L(\frac12)$$ $$\approx$$ $$1.226501376$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
good7 $$1 - 4iT - 343T^{2}$$
11 $$1 + 72T + 1.33e3T^{2}$$
13 $$1 - 6iT - 2.19e3T^{2}$$
17 $$1 + 38iT - 4.91e3T^{2}$$
19 $$1 + 52T + 6.85e3T^{2}$$
23 $$1 - 152iT - 1.21e4T^{2}$$
29 $$1 + 78T + 2.43e4T^{2}$$
31 $$1 - 120T + 2.97e4T^{2}$$
37 $$1 + 150iT - 5.06e4T^{2}$$
41 $$1 + 362T + 6.89e4T^{2}$$
43 $$1 - 484iT - 7.95e4T^{2}$$
47 $$1 + 280iT - 1.03e5T^{2}$$
53 $$1 + 670iT - 1.48e5T^{2}$$
59 $$1 - 696T + 2.05e5T^{2}$$
61 $$1 - 222T + 2.26e5T^{2}$$
67 $$1 + 4iT - 3.00e5T^{2}$$
71 $$1 + 96T + 3.57e5T^{2}$$
73 $$1 + 178iT - 3.89e5T^{2}$$
79 $$1 - 632T + 4.93e5T^{2}$$
83 $$1 + 612iT - 5.71e5T^{2}$$
89 $$1 - 994T + 7.04e5T^{2}$$
97 $$1 - 1.63e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$