# Properties

 Label 2-60-4.3-c6-0-20 Degree $2$ Conductor $60$ Sign $0.0317 + 0.999i$ 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 + (7.99 − 0.126i)2-s − 15.5i·3-s + (63.9 − 2.02i)4-s − 55.9·5-s + (−1.97 − 124. i)6-s − 335. i·7-s + (511. − 24.3i)8-s − 243·9-s + (−447. + 7.09i)10-s − 1.64e3i·11-s + (−31.6 − 997. i)12-s + 1.20e3·13-s + (−42.5 − 2.68e3i)14-s + 871. i·15-s + (4.08e3 − 259. i)16-s − 3.94e3·17-s + ⋯
 L(s)  = 1 + (0.999 − 0.0158i)2-s − 0.577i·3-s + (0.999 − 0.0317i)4-s − 0.447·5-s + (−0.00915 − 0.577i)6-s − 0.977i·7-s + (0.998 − 0.0475i)8-s − 0.333·9-s + (−0.447 + 0.00709i)10-s − 1.23i·11-s + (−0.0183 − 0.577i)12-s + 0.546·13-s + (−0.0154 − 0.977i)14-s + 0.258i·15-s + (0.997 − 0.0633i)16-s − 0.802·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0317 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.0317 + 0.999i$ 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: $$(2,\ 60,\ (\ :3),\ 0.0317 + 0.999i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$2.18210 - 2.11396i$$ $$L(\frac12)$$ $$\approx$$ $$2.18210 - 2.11396i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-7.99 + 0.126i)T$$
3 $$1 + 15.5iT$$
5 $$1 + 55.9T$$
good7 $$1 + 335. iT - 1.17e5T^{2}$$
11 $$1 + 1.64e3iT - 1.77e6T^{2}$$
13 $$1 - 1.20e3T + 4.82e6T^{2}$$
17 $$1 + 3.94e3T + 2.41e7T^{2}$$
19 $$1 + 3.73e3iT - 4.70e7T^{2}$$
23 $$1 + 4.91e3iT - 1.48e8T^{2}$$
29 $$1 - 2.03e4T + 5.94e8T^{2}$$
31 $$1 - 3.84e4iT - 8.87e8T^{2}$$
37 $$1 - 1.76e4T + 2.56e9T^{2}$$
41 $$1 - 5.41e4T + 4.75e9T^{2}$$
43 $$1 - 3.92e3iT - 6.32e9T^{2}$$
47 $$1 - 1.45e5iT - 1.07e10T^{2}$$
53 $$1 + 2.23e5T + 2.21e10T^{2}$$
59 $$1 - 3.49e5iT - 4.21e10T^{2}$$
61 $$1 - 1.44e5T + 5.15e10T^{2}$$
67 $$1 - 5.24e5iT - 9.04e10T^{2}$$
71 $$1 + 2.02e5iT - 1.28e11T^{2}$$
73 $$1 - 6.74e5T + 1.51e11T^{2}$$
79 $$1 + 3.11e5iT - 2.43e11T^{2}$$
83 $$1 + 9.75e5iT - 3.26e11T^{2}$$
89 $$1 + 1.14e6T + 4.96e11T^{2}$$
97 $$1 - 5.73e5T + 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

−13.61733932934878696474574679492, −12.68148886879937929195740257880, −11.36122370736708078475470316146, −10.70520126647119557493703219470, −8.473618437404801165560298331077, −7.19127745735602909237306420349, −6.14699197501182577318223939856, −4.42493001168466585704836851536, −3.04142554305893918843183635446, −0.958877769104565423851515474109, 2.26160125933684138518343104100, 3.90126134201507879841274643442, 5.09402583588512633037767855107, 6.46148897093615043959983619157, 8.033798969023613558435375027476, 9.597140269525392595453907550713, 11.01105647616331757758704887209, 11.97748296904117729876409344455, 12.89895379209385315504808854464, 14.29242109319576329412779374871