# Properties

 Label 2-448-448.181-c0-0-0 Degree $2$ Conductor $448$ Sign $-0.290 + 0.956i$ Analytic cond. $0.223581$ Root an. cond. $0.472843$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)8-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 + 0.707i)14-s + i·16-s − i·18-s + (−1.38 − 0.923i)22-s + (0.707 + 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.382 + 0.923i)28-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯
 L(s)  = 1 + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (−0.923 − 0.382i)7-s + (−0.923 + 0.382i)8-s + (0.923 − 0.382i)9-s + (0.324 − 1.63i)11-s + (−0.707 + 0.707i)14-s + i·16-s − i·18-s + (−1.38 − 0.923i)22-s + (0.707 + 1.70i)23-s + (−0.382 + 0.923i)25-s + (0.382 + 0.923i)28-s + (0.216 + 1.08i)29-s + (0.923 + 0.382i)32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$448$$    =    $$2^{6} \cdot 7$$ Sign: $-0.290 + 0.956i$ Analytic conductor: $$0.223581$$ Root analytic conductor: $$0.472843$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{448} (181, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 448,\ (\ :0),\ -0.290 + 0.956i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9153627997$$ $$L(\frac12)$$ $$\approx$$ $$0.9153627997$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.382 + 0.923i)T$$
7 $$1 + (0.923 + 0.382i)T$$
good3 $$1 + (-0.923 + 0.382i)T^{2}$$
5 $$1 + (0.382 - 0.923i)T^{2}$$
11 $$1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2}$$
13 $$1 + (0.382 + 0.923i)T^{2}$$
17 $$1 + iT^{2}$$
19 $$1 + (-0.382 - 0.923i)T^{2}$$
23 $$1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2}$$
29 $$1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2}$$
41 $$1 + (0.707 - 0.707i)T^{2}$$
43 $$1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2}$$
47 $$1 + iT^{2}$$
53 $$1 + (-0.382 + 1.92i)T + (-0.923 - 0.382i)T^{2}$$
59 $$1 + (0.382 - 0.923i)T^{2}$$
61 $$1 + (-0.923 + 0.382i)T^{2}$$
67 $$1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2}$$
71 $$1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2}$$
73 $$1 + (-0.707 + 0.707i)T^{2}$$
79 $$1 + (0.541 + 0.541i)T + iT^{2}$$
83 $$1 + (-0.382 - 0.923i)T^{2}$$
89 $$1 + (0.707 + 0.707i)T^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$