# Properties

 Label 1-15-15.8-r0-0-0 Degree $1$ Conductor $15$ Sign $0.525 + 0.850i$ Analytic cond. $0.0696597$ Root an. cond. $0.0696597$ 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·7-s − i·8-s − 11-s + i·13-s + 14-s + 16-s + i·17-s − 19-s − i·22-s − i·23-s − 26-s + i·28-s + 29-s + ⋯
 L(s)  = 1 + i·2-s − 4-s − i·7-s − i·8-s − 11-s + i·13-s + 14-s + 16-s + i·17-s − 19-s − i·22-s − i·23-s − 26-s + i·28-s + 29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$15$$    =    $$3 \cdot 5$$ Sign: $0.525 + 0.850i$ Analytic conductor: $$0.0696597$$ Root analytic conductor: $$0.0696597$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{15} (8, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 15,\ (0:\ ),\ 0.525 + 0.850i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4818861612 + 0.2686691313i$$ $$L(\frac12)$$ $$\approx$$ $$0.4818861612 + 0.2686691313i$$ $$L(1)$$ $$\approx$$ $$0.7385666321 + 0.3168026445i$$ $$L(1)$$ $$\approx$$ $$0.7385666321 + 0.3168026445i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1$$
7 $$1$$
11 $$1 - T$$
13 $$1$$
17 $$1$$
19 $$1 - iT$$
23 $$1 - iT$$
29 $$1$$
31 $$1$$
37 $$1 - T$$
41 $$1$$
43 $$1 + iT$$
47 $$1 + T$$
53 $$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}$$