# Properties

 Degree $2$ Conductor $450$ Sign $-0.894 + 0.447i$ Motivic weight $5$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s − 16·4-s + 118i·7-s − 64i·8-s − 192·11-s + 1.10e3i·13-s − 472·14-s + 256·16-s + 762i·17-s + 2.74e3·19-s − 768i·22-s − 1.56e3i·23-s − 4.42e3·26-s − 1.88e3i·28-s + 5.91e3·29-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 0.910i·7-s − 0.353i·8-s − 0.478·11-s + 1.81i·13-s − 0.643·14-s + 0.250·16-s + 0.639i·17-s + 1.74·19-s − 0.338i·22-s − 0.617i·23-s − 1.28·26-s − 0.455i·28-s + 1.30·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ Sign: $-0.894 + 0.447i$ Motivic weight: $$5$$ Character: $\chi_{450} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 450,\ (\ :5/2),\ -0.894 + 0.447i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$1.195334073$$ $$L(\frac12)$$ $$\approx$$ $$1.195334073$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - 4iT$$
3 $$1$$
5 $$1$$
good7 $$1 - 118iT - 1.68e4T^{2}$$
11 $$1 + 192T + 1.61e5T^{2}$$
13 $$1 - 1.10e3iT - 3.71e5T^{2}$$
17 $$1 - 762iT - 1.41e6T^{2}$$
19 $$1 - 2.74e3T + 2.47e6T^{2}$$
23 $$1 + 1.56e3iT - 6.43e6T^{2}$$
29 $$1 - 5.91e3T + 2.05e7T^{2}$$
31 $$1 + 6.86e3T + 2.86e7T^{2}$$
37 $$1 - 5.51e3iT - 6.93e7T^{2}$$
41 $$1 - 378T + 1.15e8T^{2}$$
43 $$1 + 2.43e3iT - 1.47e8T^{2}$$
47 $$1 - 1.31e4iT - 2.29e8T^{2}$$
53 $$1 - 9.17e3iT - 4.18e8T^{2}$$
59 $$1 + 3.49e4T + 7.14e8T^{2}$$
61 $$1 + 9.83e3T + 8.44e8T^{2}$$
67 $$1 + 3.37e4iT - 1.35e9T^{2}$$
71 $$1 + 7.02e4T + 1.80e9T^{2}$$
73 $$1 - 2.19e4iT - 2.07e9T^{2}$$
79 $$1 + 4.52e3T + 3.07e9T^{2}$$
83 $$1 - 1.09e5iT - 3.93e9T^{2}$$
89 $$1 - 3.84e4T + 5.58e9T^{2}$$
97 $$1 - 1.91e3iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$