# Properties

 Label 2-180-20.19-c2-0-15 Degree $2$ Conductor $180$ Sign $-0.0450 + 0.998i$ 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 + (−0.656 − 1.88i)2-s + (−3.13 + 2.48i)4-s + (3.27 − 3.77i)5-s + 9.55·7-s + (6.74 + 4.29i)8-s + (−9.28 − 3.70i)10-s + 9.92i·11-s − 7.55i·13-s + (−6.27 − 18.0i)14-s + (3.68 − 15.5i)16-s − 17.1i·17-s − 26.1i·19-s + (−0.900 + 19.9i)20-s + (18.7 − 6.51i)22-s + 1.67·23-s + ⋯
 L(s)  = 1 + (−0.328 − 0.944i)2-s + (−0.784 + 0.620i)4-s + (0.654 − 0.755i)5-s + 1.36·7-s + (0.843 + 0.537i)8-s + (−0.928 − 0.370i)10-s + 0.902i·11-s − 0.581i·13-s + (−0.448 − 1.28i)14-s + (0.230 − 0.973i)16-s − 1.01i·17-s − 1.37i·19-s + (−0.0450 + 0.998i)20-s + (0.852 − 0.296i)22-s + 0.0728·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0450 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0450 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$180$$    =    $$2^{2} \cdot 3^{2} \cdot 5$$ Sign: $-0.0450 + 0.998i$ 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: $$(2,\ 180,\ (\ :1),\ -0.0450 + 0.998i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.01264 - 1.05930i$$ $$L(\frac12)$$ $$\approx$$ $$1.01264 - 1.05930i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.656 + 1.88i)T$$
3 $$1$$
5 $$1 + (-3.27 + 3.77i)T$$
good7 $$1 - 9.55T + 49T^{2}$$
11 $$1 - 9.92iT - 121T^{2}$$
13 $$1 + 7.55iT - 169T^{2}$$
17 $$1 + 17.1iT - 289T^{2}$$
19 $$1 + 26.1iT - 361T^{2}$$
23 $$1 - 1.67T + 529T^{2}$$
29 $$1 - 0.350T + 841T^{2}$$
31 $$1 - 46.0iT - 961T^{2}$$
37 $$1 - 22.6iT - 1.36e3T^{2}$$
41 $$1 - 77.2T + 1.68e3T^{2}$$
43 $$1 + 41.7T + 1.84e3T^{2}$$
47 $$1 + 14.0T + 2.20e3T^{2}$$
53 $$1 + 22.6iT - 2.80e3T^{2}$$
59 $$1 - 94.7iT - 3.48e3T^{2}$$
61 $$1 - 38T + 3.72e3T^{2}$$
67 $$1 + 29.8T + 4.48e3T^{2}$$
71 $$1 - 7.19iT - 5.04e3T^{2}$$
73 $$1 - 34.3iT - 5.32e3T^{2}$$
79 $$1 + 46.0iT - 6.24e3T^{2}$$
83 $$1 + 24.1T + 6.88e3T^{2}$$
89 $$1 + 100.T + 7.92e3T^{2}$$
97 $$1 - 131. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$