# Properties

 Label 1-105-105.83-r1-0-0 Degree $1$ Conductor $105$ Sign $0.850 - 0.525i$ Analytic cond. $11.2838$ Root an. cond. $11.2838$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s − i·8-s − 11-s − i·13-s + 16-s − i·17-s + 19-s − i·22-s − i·23-s + 26-s + 29-s − 31-s + i·32-s + 34-s + ⋯
 L(s)  = 1 + i·2-s − 4-s − i·8-s − 11-s − i·13-s + 16-s − i·17-s + 19-s − i·22-s − i·23-s + 26-s + 29-s − 31-s + i·32-s + 34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$105$$    =    $$3 \cdot 5 \cdot 7$$ Sign: $0.850 - 0.525i$ Analytic conductor: $$11.2838$$ Root analytic conductor: $$11.2838$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{105} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 105,\ (1:\ ),\ 0.850 - 0.525i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.9751193533 - 0.2770109735i$$ $$L(\frac12)$$ $$\approx$$ $$0.9751193533 - 0.2770109735i$$ $$L(1)$$ $$\approx$$ $$0.8439644980 + 0.1992329921i$$ $$L(1)$$ $$\approx$$ $$0.8439644980 + 0.1992329921i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
7 $$1$$
good2 $$1$$
11 $$1 - T$$
13 $$1$$
17 $$1$$
19 $$1$$
23 $$1 - iT$$
29 $$1$$
31 $$1$$
37 $$1 - T$$
41 $$1$$
43 $$1 - iT$$
47 $$1$$
53 $$1$$
59 $$1 + T$$
61 $$1 - iT$$
67 $$1$$
71 $$1 + T$$
73 $$1$$
79 $$1$$
83 $$1 - iT$$
89 $$1 - iT$$
97 $$1$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$