# Properties

 Label 2-60-4.3-c6-0-15 Degree $2$ Conductor $60$ Sign $0.432 + 0.901i$ 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 + (−7.80 + 1.77i)2-s + 15.5i·3-s + (57.7 − 27.6i)4-s + 55.9·5-s + (−27.6 − 121. i)6-s − 264. i·7-s + (−400. + 318. i)8-s − 243·9-s + (−436. + 99.1i)10-s + 29.4i·11-s + (431. + 899. i)12-s − 2.75e3·13-s + (469. + 2.06e3i)14-s + 871. i·15-s + (2.56e3 − 3.19e3i)16-s + 1.56e3·17-s + ⋯
 L(s)  = 1 + (−0.975 + 0.221i)2-s + 0.577i·3-s + (0.901 − 0.432i)4-s + 0.447·5-s + (−0.128 − 0.562i)6-s − 0.771i·7-s + (−0.783 + 0.621i)8-s − 0.333·9-s + (−0.436 + 0.0991i)10-s + 0.0221i·11-s + (0.249 + 0.520i)12-s − 1.25·13-s + (0.171 + 0.751i)14-s + 0.258i·15-s + (0.625 − 0.780i)16-s + 0.317·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$60$$    =    $$2^{2} \cdot 3 \cdot 5$$ Sign: $0.432 + 0.901i$ 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: $$(2,\ 60,\ (\ :3),\ 0.432 + 0.901i)$$

## Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.726528 - 0.457244i$$ $$L(\frac12)$$ $$\approx$$ $$0.726528 - 0.457244i$$ $$L(4)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (7.80 - 1.77i)T$$
3 $$1 - 15.5iT$$
5 $$1 - 55.9T$$
good7 $$1 + 264. iT - 1.17e5T^{2}$$
11 $$1 - 29.4iT - 1.77e6T^{2}$$
13 $$1 + 2.75e3T + 4.82e6T^{2}$$
17 $$1 - 1.56e3T + 2.41e7T^{2}$$
19 $$1 + 7.29e3iT - 4.70e7T^{2}$$
23 $$1 + 1.68e4iT - 1.48e8T^{2}$$
29 $$1 - 3.62e4T + 5.94e8T^{2}$$
31 $$1 + 4.54e4iT - 8.87e8T^{2}$$
37 $$1 + 4.05e4T + 2.56e9T^{2}$$
41 $$1 - 5.46e4T + 4.75e9T^{2}$$
43 $$1 + 5.39e4iT - 6.32e9T^{2}$$
47 $$1 - 1.12e5iT - 1.07e10T^{2}$$
53 $$1 + 7.81e4T + 2.21e10T^{2}$$
59 $$1 + 2.67e5iT - 4.21e10T^{2}$$
61 $$1 + 3.53e5T + 5.15e10T^{2}$$
67 $$1 + 2.03e5iT - 9.04e10T^{2}$$
71 $$1 - 5.16e4iT - 1.28e11T^{2}$$
73 $$1 - 2.21e5T + 1.51e11T^{2}$$
79 $$1 + 5.40e5iT - 2.43e11T^{2}$$
83 $$1 + 2.64e5iT - 3.26e11T^{2}$$
89 $$1 - 3.14e5T + 4.96e11T^{2}$$
97 $$1 + 1.55e5T + 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

−13.97901402683787106839901431564, −12.28166847023247001613682101750, −10.87601924437612846011975171218, −10.07279875229413787406898918782, −9.127665566462310915716965104164, −7.70634599545083185000034884444, −6.46289714233821027569104237902, −4.79339796521800687730408211153, −2.56369329099459605065160762300, −0.48818548313816948172583399565, 1.50466470088990049116118040383, 2.84893829858162252134673740662, 5.60852717269921820429124070074, 6.98925535738640807495658331700, 8.179068631794185601783266818295, 9.362576881822827283725771734835, 10.39300763150273012172186350024, 11.94969235612368286941425618371, 12.44362274806837738002518732225, 14.05853752728203152981884769186