# Properties

 Label 1-104-104.83-r0-0-0 Degree $1$ Conductor $104$ Sign $0.957 + 0.289i$ Analytic cond. $0.482973$ Root an. cond. $0.482973$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s + i·5-s − i·7-s + 9-s + i·11-s + i·15-s − 17-s − i·19-s − i·21-s + 23-s − 25-s + 27-s − 29-s + i·31-s + i·33-s + ⋯
 L(s)  = 1 + 3-s + i·5-s − i·7-s + 9-s + i·11-s + i·15-s − 17-s − i·19-s − i·21-s + 23-s − 25-s + 27-s − 29-s + i·31-s + i·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$104$$    =    $$2^{3} \cdot 13$$ Sign: $0.957 + 0.289i$ Analytic conductor: $$0.482973$$ Root analytic conductor: $$0.482973$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{104} (83, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 104,\ (0:\ ),\ 0.957 + 0.289i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.353285148 + 0.2003792255i$$ $$L(\frac12)$$ $$\approx$$ $$1.353285148 + 0.2003792255i$$ $$L(1)$$ $$\approx$$ $$1.340819274 + 0.1222351394i$$ $$L(1)$$ $$\approx$$ $$1.340819274 + 0.1222351394i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
13 $$1$$
good3 $$1 + T$$
5 $$1$$
7 $$1 + T$$
11 $$1$$
17 $$1$$
19 $$1 - iT$$
23 $$1$$
29 $$1 + T$$
31 $$1$$
37 $$1 + iT$$
41 $$1$$
43 $$1$$
47 $$1$$
53 $$1 + iT$$
59 $$1$$
61 $$1 - T$$
67 $$1$$
71 $$1 - iT$$
73 $$1$$
79 $$1 - iT$$
83 $$1$$
89 $$1 + T$$
97 $$1$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$