# Properties

 Label 2-60-20.19-c2-0-6 Degree $2$ Conductor $60$ Sign $0.648 + 0.761i$ Analytic cond. $1.63488$ Root an. cond. $1.27862$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.52 − 1.29i)2-s + 1.73·3-s + (0.637 + 3.94i)4-s + (4.27 − 2.59i)5-s + (−2.63 − 2.24i)6-s + 0.837·7-s + (4.14 − 6.83i)8-s + 2.99·9-s + (−9.87 − 1.59i)10-s − 15.7i·11-s + (1.10 + 6.83i)12-s + 5.18i·13-s + (−1.27 − 1.08i)14-s + (7.40 − 4.49i)15-s + (−15.1 + 5.03i)16-s + 27.3i·17-s + ⋯
 L(s)  = 1 + (−0.761 − 0.648i)2-s + 0.577·3-s + (0.159 + 0.987i)4-s + (0.854 − 0.518i)5-s + (−0.439 − 0.374i)6-s + 0.119·7-s + (0.518 − 0.854i)8-s + 0.333·9-s + (−0.987 − 0.159i)10-s − 1.43i·11-s + (0.0920 + 0.569i)12-s + 0.398i·13-s + (−0.0910 − 0.0775i)14-s + (0.493 − 0.299i)15-s + (−0.949 + 0.314i)16-s + 1.60i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.648 + 0.761i$ Analytic conductor: $$1.63488$$ Root analytic conductor: $$1.27862$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{60} (19, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 60,\ (\ :1),\ 0.648 + 0.761i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.999888 - 0.461884i$$ $$L(\frac12)$$ $$\approx$$ $$0.999888 - 0.461884i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.52 + 1.29i)T$$
3 $$1 - 1.73T$$
5 $$1 + (-4.27 + 2.59i)T$$
good7 $$1 - 0.837T + 49T^{2}$$
11 $$1 + 15.7iT - 121T^{2}$$
13 $$1 - 5.18iT - 169T^{2}$$
17 $$1 - 27.3iT - 289T^{2}$$
19 $$1 - 17.9iT - 361T^{2}$$
23 $$1 + 19.1T + 529T^{2}$$
29 $$1 + 45.6T + 841T^{2}$$
31 $$1 - 13.6iT - 961T^{2}$$
37 $$1 + 15.5iT - 1.36e3T^{2}$$
41 $$1 - 13.2T + 1.68e3T^{2}$$
43 $$1 - 27.9T + 1.84e3T^{2}$$
47 $$1 + 55.6T + 2.20e3T^{2}$$
53 $$1 + 15.5iT - 2.80e3T^{2}$$
59 $$1 - 87.6iT - 3.48e3T^{2}$$
61 $$1 - 38T + 3.72e3T^{2}$$
67 $$1 - 92.2T + 4.48e3T^{2}$$
71 $$1 + 130. iT - 5.04e3T^{2}$$
73 $$1 - 54.7iT - 5.32e3T^{2}$$
79 $$1 + 13.6iT - 6.24e3T^{2}$$
83 $$1 - 59.0T + 6.88e3T^{2}$$
89 $$1 - 39.8T + 7.92e3T^{2}$$
97 $$1 + 168. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$