# Properties

 Degree $2$ Conductor $48$ Sign $0.730 + 0.682i$ Motivic weight $2$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.25 − 1.55i)2-s + (1.22 + 1.22i)3-s + (−0.846 − 3.90i)4-s + (0.909 + 0.909i)5-s + (3.44 − 0.368i)6-s − 0.654·7-s + (−7.14 − 3.59i)8-s + 2.99i·9-s + (2.55 − 0.273i)10-s + (−13.3 + 13.3i)11-s + (3.75 − 5.82i)12-s + (8.32 − 8.32i)13-s + (−0.822 + 1.01i)14-s + 2.22i·15-s + (−14.5 + 6.62i)16-s − 3.93·17-s + ⋯
 L(s)  = 1 + (0.627 − 0.778i)2-s + (0.408 + 0.408i)3-s + (−0.211 − 0.977i)4-s + (0.181 + 0.181i)5-s + (0.574 − 0.0614i)6-s − 0.0935·7-s + (−0.893 − 0.448i)8-s + 0.333i·9-s + (0.255 − 0.0273i)10-s + (−1.21 + 1.21i)11-s + (0.312 − 0.485i)12-s + (0.640 − 0.640i)13-s + (−0.0587 + 0.0728i)14-s + 0.148i·15-s + (−0.910 + 0.413i)16-s − 0.231·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.730 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$48$$    =    $$2^{4} \cdot 3$$ Sign: $0.730 + 0.682i$ Motivic weight: $$2$$ Character: $\chi_{48} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 48,\ (\ :1),\ 0.730 + 0.682i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.45848 - 0.575358i$$ $$L(\frac12)$$ $$\approx$$ $$1.45848 - 0.575358i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.25 + 1.55i)T$$
3 $$1 + (-1.22 - 1.22i)T$$
good5 $$1 + (-0.909 - 0.909i)T + 25iT^{2}$$
7 $$1 + 0.654T + 49T^{2}$$
11 $$1 + (13.3 - 13.3i)T - 121iT^{2}$$
13 $$1 + (-8.32 + 8.32i)T - 169iT^{2}$$
17 $$1 + 3.93T + 289T^{2}$$
19 $$1 + (-16.8 - 16.8i)T + 361iT^{2}$$
23 $$1 + 23.1T + 529T^{2}$$
29 $$1 + (-35.6 + 35.6i)T - 841iT^{2}$$
31 $$1 + 45.5iT - 961T^{2}$$
37 $$1 + (-10.1 - 10.1i)T + 1.36e3iT^{2}$$
41 $$1 - 28.4iT - 1.68e3T^{2}$$
43 $$1 + (-22.7 + 22.7i)T - 1.84e3iT^{2}$$
47 $$1 + 10.7iT - 2.20e3T^{2}$$
53 $$1 + (-41.5 - 41.5i)T + 2.80e3iT^{2}$$
59 $$1 + (21.0 - 21.0i)T - 3.48e3iT^{2}$$
61 $$1 + (68.7 - 68.7i)T - 3.72e3iT^{2}$$
67 $$1 + (-67.8 - 67.8i)T + 4.48e3iT^{2}$$
71 $$1 - 33.3T + 5.04e3T^{2}$$
73 $$1 - 18.6iT - 5.32e3T^{2}$$
79 $$1 + 6.29iT - 6.24e3T^{2}$$
83 $$1 + (72.0 + 72.0i)T + 6.88e3iT^{2}$$
89 $$1 + 10.6iT - 7.92e3T^{2}$$
97 $$1 - 143.T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$