# Properties

 Label 1-39-39.8-r0-0-0 Degree $1$ Conductor $39$ Sign $-0.289 - 0.957i$ Analytic cond. $0.181115$ Root an. cond. $0.181115$ 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·5-s − i·7-s + i·8-s − 10-s + i·11-s − 14-s + 16-s + 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s + ⋯
 L(s)  = 1 − i·2-s − 4-s − i·5-s − i·7-s + i·8-s − 10-s + i·11-s − 14-s + 16-s + 17-s + i·19-s + i·20-s + 22-s + 23-s − 25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$39$$    =    $$3 \cdot 13$$ Sign: $-0.289 - 0.957i$ Analytic conductor: $$0.181115$$ Root analytic conductor: $$0.181115$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{39} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 39,\ (0:\ ),\ -0.289 - 0.957i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4386508127 - 0.5911290131i$$ $$L(\frac12)$$ $$\approx$$ $$0.4386508127 - 0.5911290131i$$ $$L(1)$$ $$\approx$$ $$0.7122768728 - 0.5510574790i$$ $$L(1)$$ $$\approx$$ $$0.7122768728 - 0.5510574790i$$

## Euler product

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