# Properties

 Label 1-29-29.5-r0-0-0 Degree $1$ Conductor $29$ Sign $0.357 - 0.934i$ Analytic cond. $0.134675$ Root an. cond. $0.134675$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (−0.623 − 0.781i)11-s − 12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s − 17-s + ⋯
 L(s)  = 1 + (0.222 − 0.974i)2-s + (0.900 − 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (−0.900 + 0.433i)7-s + (−0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (0.900 + 0.433i)10-s + (−0.623 − 0.781i)11-s − 12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)14-s + (0.222 + 0.974i)15-s + (0.623 + 0.781i)16-s − 17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$29$$ Sign: $0.357 - 0.934i$ Analytic conductor: $$0.134675$$ Root analytic conductor: $$0.134675$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{29} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 29,\ (0:\ ),\ 0.357 - 0.934i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7180344568 - 0.4941222572i$$ $$L(\frac12)$$ $$\approx$$ $$0.7180344568 - 0.4941222572i$$ $$L(1)$$ $$\approx$$ $$0.9788579405 - 0.5013789824i$$ $$L(1)$$ $$\approx$$ $$0.9788579405 - 0.5013789824i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad29 $$1$$
good2 $$1 + (0.222 - 0.974i)T$$
3 $$1 + (0.900 - 0.433i)T$$
5 $$1 + (-0.222 + 0.974i)T$$
7 $$1 + (-0.900 + 0.433i)T$$
11 $$1 + (-0.623 - 0.781i)T$$
13 $$1 + (0.623 + 0.781i)T$$
17 $$1 - T$$
19 $$1 + (0.900 + 0.433i)T$$
23 $$1 + (-0.222 - 0.974i)T$$
31 $$1 + (0.222 - 0.974i)T$$
37 $$1 + (-0.623 + 0.781i)T$$
41 $$1 - T$$
43 $$1 + (0.222 + 0.974i)T$$
47 $$1 + (-0.623 - 0.781i)T$$
53 $$1 + (-0.222 + 0.974i)T$$
59 $$1 + T$$
61 $$1 + (0.900 - 0.433i)T$$
67 $$1 + (0.623 - 0.781i)T$$
71 $$1 + (0.623 + 0.781i)T$$
73 $$1 + (0.222 + 0.974i)T$$
79 $$1 + (-0.623 + 0.781i)T$$
83 $$1 + (-0.900 - 0.433i)T$$
89 $$1 + (0.222 - 0.974i)T$$
97 $$1 + (0.900 + 0.433i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$