# Properties

 Label 2-450-5.4-c5-0-19 Degree $2$ Conductor $450$ Sign $-0.447 - 0.894i$ Analytic cond. $72.1727$ Root an. cond. $8.49545$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4i·2-s − 16·4-s + 233i·7-s − 64i·8-s + 498·11-s − 809i·13-s − 932·14-s + 256·16-s + 1.00e3i·17-s + 1.70e3·19-s + 1.99e3i·22-s + 1.55e3i·23-s + 3.23e3·26-s − 3.72e3i·28-s + 7.83e3·29-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 1.79i·7-s − 0.353i·8-s + 1.24·11-s − 1.32i·13-s − 1.27·14-s + 0.250·16-s + 0.840i·17-s + 1.08·19-s + 0.877i·22-s + 0.612i·23-s + 0.938·26-s − 0.898i·28-s + 1.72·29-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ Sign: $-0.447 - 0.894i$ Analytic conductor: $$72.1727$$ Root analytic conductor: $$8.49545$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{450} (199, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 450,\ (\ :5/2),\ -0.447 - 0.894i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$2.284518836$$ $$L(\frac12)$$ $$\approx$$ $$2.284518836$$ $$L(\frac{7}{2})$$ 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 - 233iT - 1.68e4T^{2}$$
11 $$1 - 498T + 1.61e5T^{2}$$
13 $$1 + 809iT - 3.71e5T^{2}$$
17 $$1 - 1.00e3iT - 1.41e6T^{2}$$
19 $$1 - 1.70e3T + 2.47e6T^{2}$$
23 $$1 - 1.55e3iT - 6.43e6T^{2}$$
29 $$1 - 7.83e3T + 2.05e7T^{2}$$
31 $$1 - 977T + 2.86e7T^{2}$$
37 $$1 + 4.82e3iT - 6.93e7T^{2}$$
41 $$1 - 8.14e3T + 1.15e8T^{2}$$
43 $$1 + 1.94e4iT - 1.47e8T^{2}$$
47 $$1 + 8.41e3iT - 2.29e8T^{2}$$
53 $$1 - 1.76e4iT - 4.18e8T^{2}$$
59 $$1 - 3.59e4T + 7.14e8T^{2}$$
61 $$1 - 3.52e3T + 8.44e8T^{2}$$
67 $$1 - 5.74e4iT - 1.35e9T^{2}$$
71 $$1 - 7.54e3T + 1.80e9T^{2}$$
73 $$1 - 646iT - 2.07e9T^{2}$$
79 $$1 - 2.27e4T + 3.07e9T^{2}$$
83 $$1 - 1.15e4iT - 3.93e9T^{2}$$
89 $$1 + 7.89e4T + 5.58e9T^{2}$$
97 $$1 - 5.45e4iT - 8.58e9T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$