Properties

 Label 1-57-57.41-r0-0-0 Degree $1$ Conductor $57$ Sign $-0.174 + 0.984i$ Analytic cond. $0.264706$ Root an. cond. $0.264706$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
 L(s)  = 1 + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (0.939 + 0.342i)5-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.173 + 0.984i)10-s + (0.5 + 0.866i)11-s + (−0.766 − 0.642i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.173 − 0.984i)17-s − 20-s + (−0.766 + 0.642i)22-s + (0.939 − 0.342i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$1$$ Conductor: $$57$$    =    $$3 \cdot 19$$ Sign: $-0.174 + 0.984i$ Analytic conductor: $$0.264706$$ Root analytic conductor: $$0.264706$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{57} (41, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 57,\ (0:\ ),\ -0.174 + 0.984i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6077953449 + 0.7248225467i$$ $$L(\frac12)$$ $$\approx$$ $$0.6077953449 + 0.7248225467i$$ $$L(1)$$ $$\approx$$ $$0.8667234833 + 0.6187931938i$$ $$L(1)$$ $$\approx$$ $$0.8667234833 + 0.6187931938i$$

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
19 $$1$$
good2 $$1 + (0.173 + 0.984i)T$$
5 $$1 + (0.939 + 0.342i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
11 $$1 + (0.5 + 0.866i)T$$
13 $$1 + (-0.766 - 0.642i)T$$
17 $$1 + (-0.173 - 0.984i)T$$
23 $$1 + (0.939 - 0.342i)T$$
29 $$1 + (0.173 - 0.984i)T$$
31 $$1 + (0.5 - 0.866i)T$$
37 $$1 - T$$
41 $$1 + (0.766 - 0.642i)T$$
43 $$1 + (-0.939 - 0.342i)T$$
47 $$1 + (-0.173 + 0.984i)T$$
53 $$1 + (-0.939 + 0.342i)T$$
59 $$1 + (0.173 + 0.984i)T$$
61 $$1 + (-0.939 + 0.342i)T$$
67 $$1 + (-0.173 + 0.984i)T$$
71 $$1 + (-0.939 - 0.342i)T$$
73 $$1 + (0.766 - 0.642i)T$$
79 $$1 + (-0.766 + 0.642i)T$$
83 $$1 + (0.5 - 0.866i)T$$
89 $$1 + (0.766 + 0.642i)T$$
97 $$1 + (-0.173 - 0.984i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$