# Properties

 Label 1-13-13.8-r1-0-0 Degree $1$ Conductor $13$ Sign $0.289 + 0.957i$ Analytic cond. $1.39704$ Root an. cond. $1.39704$ 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 + 3-s − 4-s + i·5-s + i·6-s − i·7-s − i·8-s + 9-s − 10-s − i·11-s − 12-s + 14-s + i·15-s + 16-s − 17-s + i·18-s + ⋯
 L(s)  = 1 + i·2-s + 3-s − 4-s + i·5-s + i·6-s − i·7-s − i·8-s + 9-s − 10-s − i·11-s − 12-s + 14-s + i·15-s + 16-s − 17-s + i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$13$$ Sign: $0.289 + 0.957i$ Analytic conductor: $$1.39704$$ Root analytic conductor: $$1.39704$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{13} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 13,\ (1:\ ),\ 0.289 + 0.957i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.080873679 + 0.8020687656i$$ $$L(\frac12)$$ $$\approx$$ $$1.080873679 + 0.8020687656i$$ $$L(1)$$ $$\approx$$ $$1.086429434 + 0.5814393878i$$ $$L(1)$$ $$\approx$$ $$1.086429434 + 0.5814393878i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad13 $$1$$
good2 $$1$$
3 $$1 + T$$
5 $$1 + iT$$
7 $$1 + T$$
11 $$1 - T$$
17 $$1 + iT$$
19 $$1 - iT$$
23 $$1 - iT$$
29 $$1 + T$$
31 $$1 - T$$
37 $$1 - iT$$
41 $$1 - T$$
43 $$1$$
47 $$1 + T$$
53 $$1 + iT$$
59 $$1 + T$$
61 $$1 - T$$
67 $$1 + iT$$
71 $$1 + iT$$
73 $$1 - iT$$
79 $$1 - iT$$
83 $$1 + T$$
89 $$1 - T$$
97 $$1 - iT$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$