# Properties

 Degree $2$ Conductor $450$ Sign $-0.0627 + 0.998i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−1.80 + 1.31i)5-s + 2·7-s + (−0.809 + 0.587i)8-s + (0.690 + 2.12i)10-s + (1.61 − 4.97i)11-s + (−1.5 − 4.61i)13-s + (0.618 − 1.90i)14-s + (0.309 + 0.951i)16-s + (6.35 − 4.61i)17-s + (2.23 − 1.62i)19-s + 2.23·20-s + (−4.23 − 3.07i)22-s + (−1.85 + 5.70i)23-s + ⋯
 L(s)  = 1 + (0.218 − 0.672i)2-s + (−0.404 − 0.293i)4-s + (−0.809 + 0.587i)5-s + 0.755·7-s + (−0.286 + 0.207i)8-s + (0.218 + 0.672i)10-s + (0.487 − 1.50i)11-s + (−0.416 − 1.28i)13-s + (0.165 − 0.508i)14-s + (0.0772 + 0.237i)16-s + (1.54 − 1.11i)17-s + (0.512 − 0.372i)19-s + 0.499·20-s + (−0.903 − 0.656i)22-s + (−0.386 + 1.18i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ Sign: $-0.0627 + 0.998i$ Motivic weight: $$1$$ Character: $\chi_{450} (271, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 450,\ (\ :1/2),\ -0.0627 + 0.998i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.934932 - 0.995601i$$ $$L(\frac12)$$ $$\approx$$ $$0.934932 - 0.995601i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.309 + 0.951i)T$$
3 $$1$$
5 $$1 + (1.80 - 1.31i)T$$
good7 $$1 - 2T + 7T^{2}$$
11 $$1 + (-1.61 + 4.97i)T + (-8.89 - 6.46i)T^{2}$$
13 $$1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2}$$
17 $$1 + (-6.35 + 4.61i)T + (5.25 - 16.1i)T^{2}$$
19 $$1 + (-2.23 + 1.62i)T + (5.87 - 18.0i)T^{2}$$
23 $$1 + (1.85 - 5.70i)T + (-18.6 - 13.5i)T^{2}$$
29 $$1 + (1.11 + 0.812i)T + (8.96 + 27.5i)T^{2}$$
31 $$1 + (3 - 2.17i)T + (9.57 - 29.4i)T^{2}$$
37 $$1 + (0.663 + 2.04i)T + (-29.9 + 21.7i)T^{2}$$
41 $$1 + (-1.88 - 5.79i)T + (-33.1 + 24.0i)T^{2}$$
43 $$1 + 1.23T + 43T^{2}$$
47 $$1 + (-3.85 - 2.80i)T + (14.5 + 44.6i)T^{2}$$
53 $$1 + (-6.92 - 5.03i)T + (16.3 + 50.4i)T^{2}$$
59 $$1 + (2.76 + 8.50i)T + (-47.7 + 34.6i)T^{2}$$
61 $$1 + (2.73 - 8.42i)T + (-49.3 - 35.8i)T^{2}$$
67 $$1 + (-7.85 + 5.70i)T + (20.7 - 63.7i)T^{2}$$
71 $$1 + (11.4 + 8.33i)T + (21.9 + 67.5i)T^{2}$$
73 $$1 + (0.972 - 2.99i)T + (-59.0 - 42.9i)T^{2}$$
79 $$1 + (24.4 + 75.1i)T^{2}$$
83 $$1 + (-4.85 + 3.52i)T + (25.6 - 78.9i)T^{2}$$
89 $$1 + (0.427 - 1.31i)T + (-72.0 - 52.3i)T^{2}$$
97 $$1 + (-11.2 - 8.14i)T + (29.9 + 92.2i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$