# Properties

 Label 2-60-4.3-c6-0-6 Degree $2$ Conductor $60$ Sign $0.930 - 0.365i$ Analytic cond. $13.8032$ Root an. cond. $3.71527$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−6.61 − 4.50i)2-s + 15.5i·3-s + (23.4 + 59.5i)4-s − 55.9·5-s + (70.2 − 103. i)6-s − 489. i·7-s + (113. − 499. i)8-s − 243·9-s + (369. + 251. i)10-s + 1.67e3i·11-s + (−928. + 364. i)12-s − 528.·13-s + (−2.20e3 + 3.23e3i)14-s − 871. i·15-s + (−2.99e3 + 2.78e3i)16-s + 8.23e3·17-s + ⋯
 L(s)  = 1 + (−0.826 − 0.563i)2-s + 0.577i·3-s + (0.365 + 0.930i)4-s − 0.447·5-s + (0.325 − 0.477i)6-s − 1.42i·7-s + (0.221 − 0.975i)8-s − 0.333·9-s + (0.369 + 0.251i)10-s + 1.25i·11-s + (−0.537 + 0.211i)12-s − 0.240·13-s + (−0.803 + 1.17i)14-s − 0.258i·15-s + (−0.732 + 0.680i)16-s + 1.67·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.930 - 0.365i$ Analytic conductor: $$13.8032$$ Root analytic conductor: $$3.71527$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{60} (31, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 60,\ (\ :3),\ 0.930 - 0.365i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$1.01248 + 0.191840i$$ $$L(\frac12)$$ $$\approx$$ $$1.01248 + 0.191840i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (6.61 + 4.50i)T$$
3 $$1 - 15.5iT$$
5 $$1 + 55.9T$$
good7 $$1 + 489. iT - 1.17e5T^{2}$$
11 $$1 - 1.67e3iT - 1.77e6T^{2}$$
13 $$1 + 528.T + 4.82e6T^{2}$$
17 $$1 - 8.23e3T + 2.41e7T^{2}$$
19 $$1 + 3.89e3iT - 4.70e7T^{2}$$
23 $$1 - 2.31e4iT - 1.48e8T^{2}$$
29 $$1 - 4.39e4T + 5.94e8T^{2}$$
31 $$1 - 1.82e4iT - 8.87e8T^{2}$$
37 $$1 - 7.60e4T + 2.56e9T^{2}$$
41 $$1 + 7.73e4T + 4.75e9T^{2}$$
43 $$1 - 4.50e4iT - 6.32e9T^{2}$$
47 $$1 - 7.96e4iT - 1.07e10T^{2}$$
53 $$1 - 4.61e4T + 2.21e10T^{2}$$
59 $$1 - 1.22e5iT - 4.21e10T^{2}$$
61 $$1 - 1.73e5T + 5.15e10T^{2}$$
67 $$1 + 3.54e5iT - 9.04e10T^{2}$$
71 $$1 + 3.73e4iT - 1.28e11T^{2}$$
73 $$1 - 3.66e5T + 1.51e11T^{2}$$
79 $$1 - 1.10e5iT - 2.43e11T^{2}$$
83 $$1 + 3.53e5iT - 3.26e11T^{2}$$
89 $$1 + 9.65e5T + 4.96e11T^{2}$$
97 $$1 + 3.68e5T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$