# Properties

 Label 1-137-137.73-r0-0-0 Degree $1$ Conductor $137$ Sign $0.0841 - 0.996i$ Analytic cond. $0.636225$ Root an. cond. $0.636225$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + 10-s + (0.445 − 0.895i)11-s + (0.0922 − 0.995i)12-s + (−0.982 − 0.183i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.0922 − 0.995i)17-s + ⋯
 L(s)  = 1 + (−0.850 + 0.526i)2-s + (0.932 − 0.361i)3-s + (0.445 − 0.895i)4-s + (−0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (−0.982 − 0.183i)7-s + (0.0922 + 0.995i)8-s + (0.739 − 0.673i)9-s + 10-s + (0.445 − 0.895i)11-s + (0.0922 − 0.995i)12-s + (−0.982 − 0.183i)13-s + (0.932 − 0.361i)14-s + (−0.982 − 0.183i)15-s + (−0.602 − 0.798i)16-s + (0.0922 − 0.995i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$137$$ Sign: $0.0841 - 0.996i$ Analytic conductor: $$0.636225$$ Root analytic conductor: $$0.636225$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{137} (73, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 137,\ (0:\ ),\ 0.0841 - 0.996i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4819377846 - 0.4429468011i$$ $$L(\frac12)$$ $$\approx$$ $$0.4819377846 - 0.4429468011i$$ $$L(1)$$ $$\approx$$ $$0.7014835058 - 0.1796576414i$$ $$L(1)$$ $$\approx$$ $$0.7014835058 - 0.1796576414i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad137 $$1$$
good2 $$1 + (-0.850 + 0.526i)T$$
3 $$1 + (0.932 - 0.361i)T$$
5 $$1 + (-0.850 - 0.526i)T$$
7 $$1 + (-0.982 - 0.183i)T$$
11 $$1 + (0.445 - 0.895i)T$$
13 $$1 + (-0.982 - 0.183i)T$$
17 $$1 + (0.0922 - 0.995i)T$$
19 $$1 + (-0.273 + 0.961i)T$$
23 $$1 + (-0.602 - 0.798i)T$$
29 $$1 + (-0.602 - 0.798i)T$$
31 $$1 + (-0.273 - 0.961i)T$$
37 $$1 + T$$
41 $$1 + T$$
43 $$1 + (-0.273 + 0.961i)T$$
47 $$1 + (0.739 - 0.673i)T$$
53 $$1 + (-0.273 + 0.961i)T$$
59 $$1 + (0.739 - 0.673i)T$$
61 $$1 + (0.739 + 0.673i)T$$
67 $$1 + (-0.982 - 0.183i)T$$
71 $$1 + (0.445 + 0.895i)T$$
73 $$1 + (-0.982 + 0.183i)T$$
79 $$1 + (0.932 + 0.361i)T$$
83 $$1 + (0.0922 + 0.995i)T$$
89 $$1 + (-0.850 - 0.526i)T$$
97 $$1 + (0.445 - 0.895i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$