# Properties

 Label 2-560-5.4-c1-0-1 Degree $2$ Conductor $560$ Sign $-0.994 - 0.100i$ Analytic cond. $4.47162$ Root an. cond. $2.11462$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.44i·3-s + (−0.224 + 2.22i)5-s + i·7-s − 2.99·9-s − 4.89·11-s + 0.449i·13-s + (−5.44 − 0.550i)15-s + 2i·17-s + 6.44·19-s − 2.44·21-s − 6.89i·23-s + (−4.89 − i)25-s + 2.89·29-s + 0.898·31-s − 11.9i·33-s + ⋯
 L(s)  = 1 + 1.41i·3-s + (−0.100 + 0.994i)5-s + 0.377i·7-s − 0.999·9-s − 1.47·11-s + 0.124i·13-s + (−1.40 − 0.142i)15-s + 0.485i·17-s + 1.47·19-s − 0.534·21-s − 1.43i·23-s + (−0.979 − 0.200i)25-s + 0.538·29-s + 0.161·31-s − 2.08i·33-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$560$$    =    $$2^{4} \cdot 5 \cdot 7$$ Sign: $-0.994 - 0.100i$ Analytic conductor: $$4.47162$$ Root analytic conductor: $$2.11462$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{560} (449, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 560,\ (\ :1/2),\ -0.994 - 0.100i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0547101 + 1.08590i$$ $$L(\frac12)$$ $$\approx$$ $$0.0547101 + 1.08590i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (0.224 - 2.22i)T$$
7 $$1 - iT$$
good3 $$1 - 2.44iT - 3T^{2}$$
11 $$1 + 4.89T + 11T^{2}$$
13 $$1 - 0.449iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 - 6.44T + 19T^{2}$$
23 $$1 + 6.89iT - 23T^{2}$$
29 $$1 - 2.89T + 29T^{2}$$
31 $$1 - 0.898T + 31T^{2}$$
37 $$1 - 2iT - 37T^{2}$$
41 $$1 + 10.8T + 41T^{2}$$
43 $$1 - 8.89iT - 43T^{2}$$
47 $$1 - 0.898iT - 47T^{2}$$
53 $$1 - 1.10iT - 53T^{2}$$
59 $$1 + 6.44T + 59T^{2}$$
61 $$1 - 8.44T + 61T^{2}$$
67 $$1 - 8iT - 67T^{2}$$
71 $$1 - 10.8T + 71T^{2}$$
73 $$1 - 6.89iT - 73T^{2}$$
79 $$1 + 2.89T + 79T^{2}$$
83 $$1 - 2.44iT - 83T^{2}$$
89 $$1 - 10T + 89T^{2}$$
97 $$1 + 3.79iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$