# Properties

 Label 1-407-407.366-r0-0-0 Degree $1$ Conductor $407$ Sign $0.972 - 0.231i$ Analytic cond. $1.89010$ Root an. cond. $1.89010$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.615 + 0.788i)2-s + (0.559 − 0.829i)3-s + (−0.241 − 0.970i)4-s + (0.990 − 0.139i)5-s + (0.309 + 0.951i)6-s + (−0.719 + 0.694i)7-s + (0.913 + 0.406i)8-s + (−0.374 − 0.927i)9-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)12-s + (0.848 − 0.529i)13-s + (−0.104 − 0.994i)14-s + (0.438 − 0.898i)15-s + (−0.882 + 0.469i)16-s + (0.848 + 0.529i)17-s + (0.961 + 0.275i)18-s + ⋯
 L(s)  = 1 + (−0.615 + 0.788i)2-s + (0.559 − 0.829i)3-s + (−0.241 − 0.970i)4-s + (0.990 − 0.139i)5-s + (0.309 + 0.951i)6-s + (−0.719 + 0.694i)7-s + (0.913 + 0.406i)8-s + (−0.374 − 0.927i)9-s + (−0.5 + 0.866i)10-s + (−0.939 − 0.342i)12-s + (0.848 − 0.529i)13-s + (−0.104 − 0.994i)14-s + (0.438 − 0.898i)15-s + (−0.882 + 0.469i)16-s + (0.848 + 0.529i)17-s + (0.961 + 0.275i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$407$$    =    $$11 \cdot 37$$ Sign: $0.972 - 0.231i$ Analytic conductor: $$1.89010$$ Root analytic conductor: $$1.89010$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{407} (366, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 407,\ (0:\ ),\ 0.972 - 0.231i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.301387921 - 0.1529728762i$$ $$L(\frac12)$$ $$\approx$$ $$1.301387921 - 0.1529728762i$$ $$L(1)$$ $$\approx$$ $$1.067109522 + 0.004309906815i$$ $$L(1)$$ $$\approx$$ $$1.067109522 + 0.004309906815i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad11 $$1$$
37 $$1$$
good2 $$1 + (-0.615 + 0.788i)T$$
3 $$1 + (0.559 - 0.829i)T$$
5 $$1 + (0.990 - 0.139i)T$$
7 $$1 + (-0.719 + 0.694i)T$$
13 $$1 + (0.848 - 0.529i)T$$
17 $$1 + (0.848 + 0.529i)T$$
19 $$1 + (0.559 - 0.829i)T$$
23 $$1 + (-0.5 + 0.866i)T$$
29 $$1 + (-0.104 + 0.994i)T$$
31 $$1 + (0.309 - 0.951i)T$$
41 $$1 + (-0.997 - 0.0697i)T$$
43 $$1 + T$$
47 $$1 + (-0.104 - 0.994i)T$$
53 $$1 + (0.990 + 0.139i)T$$
59 $$1 + (0.438 - 0.898i)T$$
61 $$1 + (-0.374 + 0.927i)T$$
67 $$1 + (0.173 - 0.984i)T$$
71 $$1 + (-0.615 - 0.788i)T$$
73 $$1 + (-0.809 - 0.587i)T$$
79 $$1 + (-0.882 - 0.469i)T$$
83 $$1 + (0.848 + 0.529i)T$$
89 $$1 + (0.173 + 0.984i)T$$
97 $$1 + (0.669 + 0.743i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$