# Properties

 Label 2-60-20.19-c2-0-7 Degree $2$ Conductor $60$ Sign $0.866 - 0.5i$ 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 + (1.73 + i)2-s + 1.73·3-s + (1.99 + 3.46i)4-s − 5i·5-s + (2.99 + 1.73i)6-s − 10.3·7-s + 7.99i·8-s + 2.99·9-s + (5 − 8.66i)10-s + 10.3i·11-s + (3.46 + 5.99i)12-s − 18i·13-s + (−18 − 10.3i)14-s − 8.66i·15-s + (−8 + 13.8i)16-s + 10i·17-s + ⋯
 L(s)  = 1 + (0.866 + 0.5i)2-s + 0.577·3-s + (0.499 + 0.866i)4-s − i·5-s + (0.499 + 0.288i)6-s − 1.48·7-s + 0.999i·8-s + 0.333·9-s + (0.5 − 0.866i)10-s + 0.944i·11-s + (0.288 + 0.499i)12-s − 1.38i·13-s + (−1.28 − 0.742i)14-s − 0.577i·15-s + (−0.5 + 0.866i)16-s + 0.588i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.866 - 0.5i$ 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: $$(2,\ 60,\ (\ :1),\ 0.866 - 0.5i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$1.85513 + 0.497081i$$ $$L(\frac12)$$ $$\approx$$ $$1.85513 + 0.497081i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-1.73 - i)T$$
3 $$1 - 1.73T$$
5 $$1 + 5iT$$
good7 $$1 + 10.3T + 49T^{2}$$
11 $$1 - 10.3iT - 121T^{2}$$
13 $$1 + 18iT - 169T^{2}$$
17 $$1 - 10iT - 289T^{2}$$
19 $$1 + 13.8iT - 361T^{2}$$
23 $$1 - 6.92T + 529T^{2}$$
29 $$1 - 36T + 841T^{2}$$
31 $$1 + 6.92iT - 961T^{2}$$
37 $$1 - 54iT - 1.36e3T^{2}$$
41 $$1 - 18T + 1.68e3T^{2}$$
43 $$1 - 20.7T + 1.84e3T^{2}$$
47 $$1 + 2.20e3T^{2}$$
53 $$1 - 26iT - 2.80e3T^{2}$$
59 $$1 - 31.1iT - 3.48e3T^{2}$$
61 $$1 + 74T + 3.72e3T^{2}$$
67 $$1 + 41.5T + 4.48e3T^{2}$$
71 $$1 + 103. iT - 5.04e3T^{2}$$
73 $$1 + 36iT - 5.32e3T^{2}$$
79 $$1 + 90.0iT - 6.24e3T^{2}$$
83 $$1 + 90.0T + 6.88e3T^{2}$$
89 $$1 - 18T + 7.92e3T^{2}$$
97 $$1 + 72iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$