# Properties

 Label 2-712-712.427-c0-0-0 Degree $2$ Conductor $712$ Sign $0.995 + 0.0899i$ Analytic cond. $0.355334$ Root an. cond. $0.596099$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.654 + 0.755i)2-s + (1.12 − 1.50i)3-s + (−0.142 + 0.989i)4-s + (1.86 − 0.133i)6-s + (−0.841 + 0.540i)8-s + (−0.707 − 2.40i)9-s + (−0.909 − 0.584i)11-s + (1.32 + 1.32i)12-s + (−0.959 − 0.281i)16-s + (0.989 + 0.857i)17-s + (1.35 − 2.11i)18-s + (−0.936 + 1.71i)19-s + (−0.153 − 1.07i)22-s + (−0.133 + 1.86i)24-s + (−0.415 + 0.909i)25-s + ⋯
 L(s)  = 1 + (0.654 + 0.755i)2-s + (1.12 − 1.50i)3-s + (−0.142 + 0.989i)4-s + (1.86 − 0.133i)6-s + (−0.841 + 0.540i)8-s + (−0.707 − 2.40i)9-s + (−0.909 − 0.584i)11-s + (1.32 + 1.32i)12-s + (−0.959 − 0.281i)16-s + (0.989 + 0.857i)17-s + (1.35 − 2.11i)18-s + (−0.936 + 1.71i)19-s + (−0.153 − 1.07i)22-s + (−0.133 + 1.86i)24-s + (−0.415 + 0.909i)25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$712$$    =    $$2^{3} \cdot 89$$ Sign: $0.995 + 0.0899i$ Analytic conductor: $$0.355334$$ Root analytic conductor: $$0.596099$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{712} (427, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 712,\ (\ :0),\ 0.995 + 0.0899i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.660114731$$ $$L(\frac12)$$ $$\approx$$ $$1.660114731$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.654 - 0.755i)T$$
89 $$1 + (0.415 + 0.909i)T$$
good3 $$1 + (-1.12 + 1.50i)T + (-0.281 - 0.959i)T^{2}$$
5 $$1 + (0.415 - 0.909i)T^{2}$$
7 $$1 + (-0.909 - 0.415i)T^{2}$$
11 $$1 + (0.909 + 0.584i)T + (0.415 + 0.909i)T^{2}$$
13 $$1 + (-0.281 - 0.959i)T^{2}$$
17 $$1 + (-0.989 - 0.857i)T + (0.142 + 0.989i)T^{2}$$
19 $$1 + (0.936 - 1.71i)T + (-0.540 - 0.841i)T^{2}$$
23 $$1 + (-0.540 - 0.841i)T^{2}$$
29 $$1 + (0.909 + 0.415i)T^{2}$$
31 $$1 + (-0.540 + 0.841i)T^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + (-1.13 + 0.847i)T + (0.281 - 0.959i)T^{2}$$
43 $$1 + (0.0303 + 0.139i)T + (-0.909 + 0.415i)T^{2}$$
47 $$1 + (-0.959 - 0.281i)T^{2}$$
53 $$1 + (-0.959 + 0.281i)T^{2}$$
59 $$1 + (1.19 + 1.59i)T + (-0.281 + 0.959i)T^{2}$$
61 $$1 + (0.755 + 0.654i)T^{2}$$
67 $$1 + (0.0801 + 0.557i)T + (-0.959 + 0.281i)T^{2}$$
71 $$1 + (0.415 + 0.909i)T^{2}$$
73 $$1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2}$$
79 $$1 + (0.841 + 0.540i)T^{2}$$
83 $$1 + (-1.19 + 0.0855i)T + (0.989 - 0.142i)T^{2}$$
97 $$1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$