# Properties

 Label 1-448-448.69-r1-0-0 Degree $1$ Conductor $448$ Sign $0.0980 - 0.995i$ Analytic cond. $48.1442$ Root an. cond. $48.1442$ 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)3-s + (−0.923 − 0.382i)5-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s + i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 − 0.382i)37-s + ⋯
 L(s)  = 1 + (−0.382 − 0.923i)3-s + (−0.923 − 0.382i)5-s + (−0.707 + 0.707i)9-s + (−0.382 + 0.923i)11-s + (−0.923 + 0.382i)13-s + i·15-s − i·17-s + (0.923 − 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.382 − 0.923i)29-s + 31-s + 33-s + (−0.923 − 0.382i)37-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$448$$    =    $$2^{6} \cdot 7$$ Sign: $0.0980 - 0.995i$ Analytic conductor: $$48.1442$$ Root analytic conductor: $$48.1442$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{448} (69, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 448,\ (1:\ ),\ 0.0980 - 0.995i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6579554681 - 0.5963360759i$$ $$L(\frac12)$$ $$\approx$$ $$0.6579554681 - 0.5963360759i$$ $$L(1)$$ $$\approx$$ $$0.6894468168 - 0.2036009330i$$ $$L(1)$$ $$\approx$$ $$0.6894468168 - 0.2036009330i$$

## Euler product

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