# Properties

 Label 1-37-37.6-r1-0-0 Degree $1$ Conductor $37$ Sign $0.763 + 0.646i$ 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 − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 7-s + i·8-s + 9-s + 10-s − 11-s + 12-s + i·13-s − i·14-s − i·15-s + 16-s + i·17-s + ⋯
 L(s)  = 1 − i·2-s − 3-s − 4-s + i·5-s + i·6-s + 7-s + i·8-s + 9-s + 10-s − 11-s + 12-s + i·13-s − i·14-s − i·15-s + 16-s + i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$37$$ Sign: $0.763 + 0.646i$ Analytic conductor: $$3.97620$$ Root analytic conductor: $$3.97620$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{37} (6, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 37,\ (1:\ ),\ 0.763 + 0.646i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7412395992 + 0.2717602033i$$ $$L(\frac12)$$ $$\approx$$ $$0.7412395992 + 0.2717602033i$$ $$L(1)$$ $$\approx$$ $$0.7279167277 - 0.06024423023i$$ $$L(1)$$ $$\approx$$ $$0.7279167277 - 0.06024423023i$$

## Euler product

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