# Properties

 Label 1-3381-3381.536-r0-0-0 Degree $1$ Conductor $3381$ Sign $-0.338 - 0.940i$ Analytic cond. $15.7012$ Root an. cond. $15.7012$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.568 − 0.822i)2-s + (−0.352 − 0.935i)4-s + (−0.942 − 0.333i)5-s + (−0.970 − 0.242i)8-s + (−0.810 + 0.585i)10-s + (0.155 + 0.987i)11-s + (−0.714 + 0.699i)13-s + (−0.751 + 0.659i)16-s + (−0.634 − 0.773i)17-s + (−0.235 + 0.971i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (0.777 + 0.628i)25-s + (0.169 + 0.985i)26-s + (−0.986 + 0.162i)29-s + ⋯
 L(s)  = 1 + (0.568 − 0.822i)2-s + (−0.352 − 0.935i)4-s + (−0.942 − 0.333i)5-s + (−0.970 − 0.242i)8-s + (−0.810 + 0.585i)10-s + (0.155 + 0.987i)11-s + (−0.714 + 0.699i)13-s + (−0.751 + 0.659i)16-s + (−0.634 − 0.773i)17-s + (−0.235 + 0.971i)19-s + (0.0203 + 0.999i)20-s + (0.900 + 0.433i)22-s + (0.777 + 0.628i)25-s + (0.169 + 0.985i)26-s + (−0.986 + 0.162i)29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.338 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$3381$$    =    $$3 \cdot 7^{2} \cdot 23$$ Sign: $-0.338 - 0.940i$ Analytic conductor: $$15.7012$$ Root analytic conductor: $$15.7012$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3381} (536, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 3381,\ (0:\ ),\ -0.338 - 0.940i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6766956850 - 0.9628358810i$$ $$L(\frac12)$$ $$\approx$$ $$0.6766956850 - 0.9628358810i$$ $$L(1)$$ $$\approx$$ $$0.8580263501 - 0.4832098662i$$ $$L(1)$$ $$\approx$$ $$0.8580263501 - 0.4832098662i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
23 $$1$$
good2 $$1 + (0.568 - 0.822i)T$$
5 $$1 + (-0.942 - 0.333i)T$$
11 $$1 + (0.155 + 0.987i)T$$
13 $$1 + (-0.714 + 0.699i)T$$
17 $$1 + (-0.634 - 0.773i)T$$
19 $$1 + (-0.235 + 0.971i)T$$
29 $$1 + (-0.986 + 0.162i)T$$
31 $$1 + (0.580 - 0.814i)T$$
37 $$1 + (0.802 - 0.596i)T$$
41 $$1 + (-0.182 - 0.983i)T$$
43 $$1 + (-0.970 + 0.242i)T$$
47 $$1 + (0.733 + 0.680i)T$$
53 $$1 + (0.601 + 0.798i)T$$
59 $$1 + (-0.810 + 0.585i)T$$
61 $$1 + (-0.938 + 0.346i)T$$
67 $$1 + (0.786 - 0.618i)T$$
71 $$1 + (0.301 - 0.953i)T$$
73 $$1 + (0.511 - 0.859i)T$$
79 $$1 + (0.888 - 0.458i)T$$
83 $$1 + (-0.591 + 0.806i)T$$
89 $$1 + (0.665 - 0.746i)T$$
97 $$1 + (0.654 - 0.755i)T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$