# Properties

 Label 1-155-155.99-r1-0-0 Degree $1$ Conductor $155$ Sign $-0.970 - 0.242i$ Analytic cond. $16.6570$ Root an. cond. $16.6570$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
 L(s)  = 1 − 2-s + (−0.5 + 0.866i)3-s + 4-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$155$$    =    $$5 \cdot 31$$ Sign: $-0.970 - 0.242i$ Analytic conductor: $$16.6570$$ Root analytic conductor: $$16.6570$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{155} (99, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 155,\ (1:\ ),\ -0.970 - 0.242i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.01767066748 + 0.1435588858i$$ $$L(\frac12)$$ $$\approx$$ $$0.01767066748 + 0.1435588858i$$ $$L(1)$$ $$\approx$$ $$0.4896129383 + 0.1223872502i$$ $$L(1)$$ $$\approx$$ $$0.4896129383 + 0.1223872502i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
31 $$1$$
good2 $$1 - T$$
3 $$1 + (-0.5 + 0.866i)T$$
7 $$1 + (0.5 - 0.866i)T$$
11 $$1 + (0.5 + 0.866i)T$$
13 $$1 + (-0.5 - 0.866i)T$$
17 $$1 + (-0.5 + 0.866i)T$$
19 $$1 + (-0.5 + 0.866i)T$$
23 $$1 + T$$
29 $$1 - T$$
37 $$1 + (-0.5 + 0.866i)T$$
41 $$1 + (-0.5 - 0.866i)T$$
43 $$1 + (-0.5 + 0.866i)T$$
47 $$1 - T$$
53 $$1 + (-0.5 - 0.866i)T$$
59 $$1 + (-0.5 + 0.866i)T$$
61 $$1 - T$$
67 $$1 + (0.5 + 0.866i)T$$
71 $$1 + (-0.5 - 0.866i)T$$
73 $$1 + (-0.5 - 0.866i)T$$
79 $$1 + (0.5 - 0.866i)T$$
83 $$1 + (-0.5 - 0.866i)T$$
89 $$1 - T$$
97 $$1 - T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$