# Properties

 Label 1-185-185.43-r0-0-0 Degree $1$ Conductor $185$ Sign $-0.988 - 0.148i$ Analytic cond. $0.859136$ Root an. cond. $0.859136$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

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

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$185$$    =    $$5 \cdot 37$$ Sign: $-0.988 - 0.148i$ Analytic conductor: $$0.859136$$ Root analytic conductor: $$0.859136$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{185} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 185,\ (0:\ ),\ -0.988 - 0.148i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.03006053732 - 0.4020746814i$$ $$L(\frac12)$$ $$\approx$$ $$0.03006053732 - 0.4020746814i$$ $$L(1)$$ $$\approx$$ $$0.4523173885 - 0.2953422780i$$ $$L(1)$$ $$\approx$$ $$0.4523173885 - 0.2953422780i$$

## Euler product

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