# Properties

 Label 1-185-185.83-r1-0-0 Degree $1$ Conductor $185$ Sign $0.863 + 0.504i$ Analytic cond. $19.8810$ Root an. cond. $19.8810$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.342 + 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (0.984 − 0.173i)7-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯
 L(s)  = 1 + (0.342 + 0.939i)2-s + (0.342 − 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (0.984 − 0.173i)7-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.342 + 0.939i)12-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + (0.342 − 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.863 + 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$185$$    =    $$5 \cdot 37$$ Sign: $0.863 + 0.504i$ Analytic conductor: $$19.8810$$ Root analytic conductor: $$19.8810$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{185} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 185,\ (1:\ ),\ 0.863 + 0.504i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.460326105 + 0.6668105238i$$ $$L(\frac12)$$ $$\approx$$ $$2.460326105 + 0.6668105238i$$ $$L(1)$$ $$\approx$$ $$1.470846577 + 0.3394072866i$$ $$L(1)$$ $$\approx$$ $$1.470846577 + 0.3394072866i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
37 $$1$$
good2 $$1 + (0.342 + 0.939i)T$$
3 $$1 + (0.342 - 0.939i)T$$
7 $$1 + (0.984 - 0.173i)T$$
11 $$1 + (-0.5 + 0.866i)T$$
13 $$1 + (0.642 + 0.766i)T$$
17 $$1 + (0.642 - 0.766i)T$$
19 $$1 + (0.939 + 0.342i)T$$
23 $$1 + (0.866 - 0.5i)T$$
29 $$1 + (0.5 - 0.866i)T$$
31 $$1 + T$$
41 $$1 + (0.766 - 0.642i)T$$
43 $$1 + iT$$
47 $$1 + (-0.866 + 0.5i)T$$
53 $$1 + (0.984 + 0.173i)T$$
59 $$1 + (-0.173 + 0.984i)T$$
61 $$1 + (0.766 - 0.642i)T$$
67 $$1 + (0.984 - 0.173i)T$$
71 $$1 + (-0.939 - 0.342i)T$$
73 $$1 + iT$$
79 $$1 + (-0.173 - 0.984i)T$$
83 $$1 + (-0.642 + 0.766i)T$$
89 $$1 + (-0.173 + 0.984i)T$$
97 $$1 + (-0.866 + 0.5i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$