# Properties

 Label 2-1800-5.4-c3-0-60 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 − 2i·7-s + 34·11-s − 68i·13-s − 38i·17-s − 4·19-s − 152i·23-s − 46·29-s − 260·31-s + 312i·37-s − 48·41-s − 200i·43-s + 104i·47-s + 339·49-s + 414i·53-s − 2·59-s + ⋯
 L(s)  = 1 − 0.107i·7-s + 0.931·11-s − 1.45i·13-s − 0.542i·17-s − 0.0482·19-s − 1.37i·23-s − 0.294·29-s − 1.50·31-s + 1.38i·37-s − 0.182·41-s − 0.709i·43-s + 0.322i·47-s + 0.988·49-s + 1.07i·53-s − 0.00441·59-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

 $$L(2)$$ $$\approx$$ $$1.046462934$$ $$L(\frac12)$$ $$\approx$$ $$1.046462934$$ $$L(\frac{5}{2})$$ 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 + 2iT - 343T^{2}$$
11 $$1 - 34T + 1.33e3T^{2}$$
13 $$1 + 68iT - 2.19e3T^{2}$$
17 $$1 + 38iT - 4.91e3T^{2}$$
19 $$1 + 4T + 6.85e3T^{2}$$
23 $$1 + 152iT - 1.21e4T^{2}$$
29 $$1 + 46T + 2.43e4T^{2}$$
31 $$1 + 260T + 2.97e4T^{2}$$
37 $$1 - 312iT - 5.06e4T^{2}$$
41 $$1 + 48T + 6.89e4T^{2}$$
43 $$1 + 200iT - 7.95e4T^{2}$$
47 $$1 - 104iT - 1.03e5T^{2}$$
53 $$1 - 414iT - 1.48e5T^{2}$$
59 $$1 + 2T + 2.05e5T^{2}$$
61 $$1 + 38T + 2.26e5T^{2}$$
67 $$1 - 244iT - 3.00e5T^{2}$$
71 $$1 + 708T + 3.57e5T^{2}$$
73 $$1 + 378iT - 3.89e5T^{2}$$
79 $$1 - 852T + 4.93e5T^{2}$$
83 $$1 + 844iT - 5.71e5T^{2}$$
89 $$1 + 1.38e3T + 7.04e5T^{2}$$
97 $$1 + 514iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$