# Properties

 Label 2-180-20.19-c2-0-4 Degree $2$ Conductor $180$ Sign $-0.484 - 0.875i$ Analytic cond. $4.90464$ Root an. cond. $2.21464$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.93 + 0.5i)2-s + (3.50 − 1.93i)4-s + 5i·5-s + (−5.80 + 5.50i)8-s + (−2.5 − 9.68i)10-s + (8.50 − 13.5i)16-s + 14i·17-s + 30.9i·19-s + (9.68 + 17.5i)20-s − 30.9·23-s − 25·25-s + 61.9i·31-s + (−9.68 + 30.5i)32-s + (−7 − 27.1i)34-s + (−15.4 − 60.0i)38-s + ⋯
 L(s)  = 1 + (−0.968 + 0.250i)2-s + (0.875 − 0.484i)4-s + i·5-s + (−0.726 + 0.687i)8-s + (−0.250 − 0.968i)10-s + (0.531 − 0.847i)16-s + 0.823i·17-s + 1.63i·19-s + (0.484 + 0.875i)20-s − 1.34·23-s − 25-s + 1.99i·31-s + (−0.302 + 0.953i)32-s + (−0.205 − 0.797i)34-s + (−0.407 − 1.57i)38-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.484 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$180$$    =    $$2^{2} \cdot 3^{2} \cdot 5$$ Sign: $-0.484 - 0.875i$ Analytic conductor: $$4.90464$$ Root analytic conductor: $$2.21464$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{180} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 180,\ (\ :1),\ -0.484 - 0.875i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.380434 + 0.645269i$$ $$L(\frac12)$$ $$\approx$$ $$0.380434 + 0.645269i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.93 - 0.5i)T$$
3 $$1$$
5 $$1 - 5iT$$
good7 $$1 + 49T^{2}$$
11 $$1 - 121T^{2}$$
13 $$1 - 169T^{2}$$
17 $$1 - 14iT - 289T^{2}$$
19 $$1 - 30.9iT - 361T^{2}$$
23 $$1 + 30.9T + 529T^{2}$$
29 $$1 + 841T^{2}$$
31 $$1 - 61.9iT - 961T^{2}$$
37 $$1 - 1.36e3T^{2}$$
41 $$1 + 1.68e3T^{2}$$
43 $$1 + 1.84e3T^{2}$$
47 $$1 - 92.9T + 2.20e3T^{2}$$
53 $$1 + 86iT - 2.80e3T^{2}$$
59 $$1 - 3.48e3T^{2}$$
61 $$1 - 118T + 3.72e3T^{2}$$
67 $$1 + 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 5.32e3T^{2}$$
79 $$1 + 123. iT - 6.24e3T^{2}$$
83 $$1 + 61.9T + 6.88e3T^{2}$$
89 $$1 + 7.92e3T^{2}$$
97 $$1 - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$