# Properties

 Label 2-360-120.59-c1-0-23 Degree $2$ Conductor $360$ Sign $-0.577 - 0.816i$ Analytic cond. $2.87461$ Root an. cond. $1.69546$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s − 2.00·4-s − 2.23i·5-s − 5.16·7-s + 2.82i·8-s − 3.16·10-s + 3.05i·11-s + 0.837·13-s + 7.30i·14-s + 4.00·16-s − 6.32·19-s + 4.47i·20-s + 4.32·22-s − 4.47i·23-s − 5.00·25-s − 1.18i·26-s + ⋯
 L(s)  = 1 − 0.999i·2-s − 1.00·4-s − 0.999i·5-s − 1.95·7-s + 1.00i·8-s − 1.00·10-s + 0.921i·11-s + 0.232·13-s + 1.95i·14-s + 1.00·16-s − 1.45·19-s + 1.00i·20-s + 0.921·22-s − 0.932i·23-s − 1.00·25-s − 0.232i·26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$360$$    =    $$2^{3} \cdot 3^{2} \cdot 5$$ Sign: $-0.577 - 0.816i$ Analytic conductor: $$2.87461$$ Root analytic conductor: $$1.69546$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{360} (179, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 360,\ (\ :1/2),\ -0.577 - 0.816i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.124369 + 0.240263i$$ $$L(\frac12)$$ $$\approx$$ $$0.124369 + 0.240263i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41iT$$
3 $$1$$
5 $$1 + 2.23iT$$
good7 $$1 + 5.16T + 7T^{2}$$
11 $$1 - 3.05iT - 11T^{2}$$
13 $$1 - 0.837T + 13T^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + 6.32T + 19T^{2}$$
23 $$1 + 4.47iT - 23T^{2}$$
29 $$1 + 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 + 11.1T + 37T^{2}$$
41 $$1 + 10.3iT - 41T^{2}$$
43 $$1 - 43T^{2}$$
47 $$1 + 2.82iT - 47T^{2}$$
53 $$1 - 5.65iT - 53T^{2}$$
59 $$1 + 5.42iT - 59T^{2}$$
61 $$1 - 61T^{2}$$
67 $$1 - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 - 73T^{2}$$
79 $$1 - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 + 18.8iT - 89T^{2}$$
97 $$1 - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$