# Properties

 Label 2-712-712.411-c0-0-0 Degree $2$ Conductor $712$ Sign $0.484 + 0.874i$ 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 − 2-s + (1 − i)3-s + 4-s + (−1 + i)6-s − 8-s − i·9-s + (1 − i)12-s + 16-s − 2i·17-s + i·18-s + (1 + i)19-s + (−1 + i)24-s − 25-s − 32-s + 2i·34-s + ⋯
 L(s)  = 1 − 2-s + (1 − i)3-s + 4-s + (−1 + i)6-s − 8-s − i·9-s + (1 − i)12-s + 16-s − 2i·17-s + i·18-s + (1 + i)19-s + (−1 + i)24-s − 25-s − 32-s + 2i·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$712$$    =    $$2^{3} \cdot 89$$ Sign: $0.484 + 0.874i$ Analytic conductor: $$0.355334$$ Root analytic conductor: $$0.596099$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{712} (411, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 712,\ (\ :0),\ 0.484 + 0.874i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8571746403$$ $$L(\frac12)$$ $$\approx$$ $$0.8571746403$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
89 $$1 + T$$
good3 $$1 + (-1 + i)T - iT^{2}$$
5 $$1 + T^{2}$$
7 $$1 + iT^{2}$$
11 $$1 + T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + 2iT - T^{2}$$
19 $$1 + (-1 - i)T + iT^{2}$$
23 $$1 + iT^{2}$$
29 $$1 - iT^{2}$$
31 $$1 - iT^{2}$$
37 $$1 - iT^{2}$$
41 $$1 + (1 - i)T - iT^{2}$$
43 $$1 + (1 - i)T - iT^{2}$$
47 $$1 + T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + (-1 - i)T + iT^{2}$$
61 $$1 + iT^{2}$$
67 $$1 + T^{2}$$
71 $$1 + T^{2}$$
73 $$1 + T^{2}$$
79 $$1 + T^{2}$$
83 $$1 + (1 - i)T - iT^{2}$$
97 $$1 - 2T + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$