# Properties

 Label 2-60-4.3-c6-0-16 Degree $2$ Conductor $60$ Sign $0.613 + 0.789i$ Analytic cond. $13.8032$ Root an. cond. $3.71527$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (2.59 + 7.56i)2-s + 15.5i·3-s + (−50.5 + 39.2i)4-s − 55.9·5-s + (−117. + 40.4i)6-s − 671. i·7-s + (−428. − 280. i)8-s − 243·9-s + (−144. − 423. i)10-s + 199. i·11-s + (−611. − 787. i)12-s + 2.04e3·13-s + (5.07e3 − 1.74e3i)14-s − 871. i·15-s + (1.01e3 − 3.96e3i)16-s − 8.21e3·17-s + ⋯
 L(s)  = 1 + (0.324 + 0.945i)2-s + 0.577i·3-s + (−0.789 + 0.613i)4-s − 0.447·5-s + (−0.546 + 0.187i)6-s − 1.95i·7-s + (−0.836 − 0.548i)8-s − 0.333·9-s + (−0.144 − 0.423i)10-s + 0.149i·11-s + (−0.354 − 0.455i)12-s + 0.929·13-s + (1.85 − 0.634i)14-s − 0.258i·15-s + (0.247 − 0.968i)16-s − 1.67·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.613 + 0.789i$ 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: $$(2,\ 60,\ (\ :3),\ 0.613 + 0.789i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.692038 - 0.338779i$$ $$L(\frac12)$$ $$\approx$$ $$0.692038 - 0.338779i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-2.59 - 7.56i)T$$
3 $$1 - 15.5iT$$
5 $$1 + 55.9T$$
good7 $$1 + 671. iT - 1.17e5T^{2}$$
11 $$1 - 199. iT - 1.77e6T^{2}$$
13 $$1 - 2.04e3T + 4.82e6T^{2}$$
17 $$1 + 8.21e3T + 2.41e7T^{2}$$
19 $$1 - 5.14e3iT - 4.70e7T^{2}$$
23 $$1 + 1.46e4iT - 1.48e8T^{2}$$
29 $$1 + 1.25e4T + 5.94e8T^{2}$$
31 $$1 + 5.44e4iT - 8.87e8T^{2}$$
37 $$1 + 11.6T + 2.56e9T^{2}$$
41 $$1 + 6.78e4T + 4.75e9T^{2}$$
43 $$1 + 1.05e5iT - 6.32e9T^{2}$$
47 $$1 - 8.96e4iT - 1.07e10T^{2}$$
53 $$1 + 1.36e3T + 2.21e10T^{2}$$
59 $$1 + 5.60e3iT - 4.21e10T^{2}$$
61 $$1 + 1.32e5T + 5.15e10T^{2}$$
67 $$1 - 3.25e5iT - 9.04e10T^{2}$$
71 $$1 - 3.03e5iT - 1.28e11T^{2}$$
73 $$1 + 2.47e5T + 1.51e11T^{2}$$
79 $$1 + 3.89e5iT - 2.43e11T^{2}$$
83 $$1 + 2.15e5iT - 3.26e11T^{2}$$
89 $$1 - 4.62e4T + 4.96e11T^{2}$$
97 $$1 - 8.00e5T + 8.32e11T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.79699808235389448604713423409, −13.05070609731510891999454656072, −11.26521212060027680143595648595, −10.19616161623005687452322071361, −8.685251421407802100738015967769, −7.50632930073835848599110188679, −6.37694994025025738577839294214, −4.44472742270473626821805483510, −3.82297276831820406617990289795, −0.28526202410606215602659275119, 1.80597965184834123350176174419, 3.12893051284227983984114338065, 5.08031935222318889165560351904, 6.34649563706348074811178066735, 8.534204295508521910653208144905, 9.132175041051086414067338960092, 11.09793606788059864314596404715, 11.73897760787068969845605301073, 12.74522647542493145969450932739, 13.62723356125158881743752545346