# Properties

 Label 1-37-37.20-r1-0-0 Degree $1$ Conductor $37$ Sign $0.849 - 0.527i$ Analytic cond. $3.97620$ Root an. cond. $3.97620$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s − i·6-s + (0.173 + 0.984i)7-s + (0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + ⋯
 L(s)  = 1 + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s − i·6-s + (0.173 + 0.984i)7-s + (0.866 + 0.5i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)12-s + (−0.642 − 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$37$$ Sign: $0.849 - 0.527i$ Analytic conductor: $$3.97620$$ Root analytic conductor: $$3.97620$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{37} (20, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 37,\ (1:\ ),\ 0.849 - 0.527i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.714553287 - 0.4889529956i$$ $$L(\frac12)$$ $$\approx$$ $$1.714553287 - 0.4889529956i$$ $$L(1)$$ $$\approx$$ $$1.300246394 - 0.3269880849i$$ $$L(1)$$ $$\approx$$ $$1.300246394 - 0.3269880849i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad37 $$1$$
good2 $$1 + (-0.342 - 0.939i)T$$
3 $$1 + (0.939 + 0.342i)T$$
5 $$1 + (0.984 - 0.173i)T$$
7 $$1 + (0.173 + 0.984i)T$$
11 $$1 + (0.5 - 0.866i)T$$
13 $$1 + (-0.642 - 0.766i)T$$
17 $$1 + (0.642 - 0.766i)T$$
19 $$1 + (-0.342 + 0.939i)T$$
23 $$1 + (-0.866 + 0.5i)T$$
29 $$1 + (-0.866 - 0.5i)T$$
31 $$1 + iT$$
41 $$1 + (-0.766 + 0.642i)T$$
43 $$1 - iT$$
47 $$1 + (-0.5 - 0.866i)T$$
53 $$1 + (0.173 - 0.984i)T$$
59 $$1 + (-0.984 - 0.173i)T$$
61 $$1 + (0.642 + 0.766i)T$$
67 $$1 + (-0.173 - 0.984i)T$$
71 $$1 + (-0.939 - 0.342i)T$$
73 $$1 - T$$
79 $$1 + (0.984 - 0.173i)T$$
83 $$1 + (0.766 + 0.642i)T$$
89 $$1 + (0.984 + 0.173i)T$$
97 $$1 + (-0.866 + 0.5i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$