# Properties

 Degree $2$ Conductor $450$ Sign $-0.939 + 0.342i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s − 1.73i·3-s + (−0.499 − 0.866i)4-s + (−1.49 − 0.866i)6-s + (1 − 1.73i)7-s − 0.999·8-s − 2.99·9-s + (−1.49 + 0.866i)12-s + (−2 − 3.46i)13-s + (−0.999 − 1.73i)14-s + (−0.5 + 0.866i)16-s + 6·17-s + (−1.49 + 2.59i)18-s − 7·19-s + (−2.99 − 1.73i)21-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s − 0.999i·3-s + (−0.249 − 0.433i)4-s + (−0.612 − 0.353i)6-s + (0.377 − 0.654i)7-s − 0.353·8-s − 0.999·9-s + (−0.433 + 0.249i)12-s + (−0.554 − 0.960i)13-s + (−0.267 − 0.462i)14-s + (−0.125 + 0.216i)16-s + 1.45·17-s + (−0.353 + 0.612i)18-s − 1.60·19-s + (−0.654 − 0.377i)21-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$450$$    =    $$2 \cdot 3^{2} \cdot 5^{2}$$ Sign: $-0.939 + 0.342i$ Motivic weight: $$1$$ Character: $\chi_{450} (151, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 450,\ (\ :1/2),\ -0.939 + 0.342i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.253086 - 1.43532i$$ $$L(\frac12)$$ $$\approx$$ $$0.253086 - 1.43532i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
3 $$1 + 1.73iT$$
5 $$1$$
good7 $$1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 - 6T + 17T^{2}$$
19 $$1 + 7T + 19T^{2}$$
23 $$1 + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 2T + 37T^{2}$$
41 $$1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 12T + 53T^{2}$$
59 $$1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 6T + 71T^{2}$$
73 $$1 - T + 73T^{2}$$
79 $$1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 - 15T + 89T^{2}$$
97 $$1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$