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Properties

Label 2-60-20.3-c3-0-8
Degree $2$
Conductor $60$
Sign $-0.454 + 0.890i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 − 1.83i)2-s + (−2.12 + 2.12i)3-s + (1.25 + 7.90i)4-s + (8.47 − 7.28i)5-s + (8.45 − 0.665i)6-s + (−13.7 − 13.7i)7-s + (11.8 − 19.2i)8-s − 8.99i·9-s + (−31.6 + 0.104i)10-s − 25.6i·11-s + (−19.4 − 14.1i)12-s + (−46.9 − 46.9i)13-s + (4.32 + 54.8i)14-s + (−2.52 + 33.4i)15-s + (−60.8 + 19.7i)16-s + (38.6 − 38.6i)17-s + ⋯
L(s)  = 1  + (−0.760 − 0.649i)2-s + (−0.408 + 0.408i)3-s + (0.156 + 0.987i)4-s + (0.758 − 0.651i)5-s + (0.575 − 0.0453i)6-s + (−0.743 − 0.743i)7-s + (0.522 − 0.852i)8-s − 0.333i·9-s + (−0.999 + 0.00330i)10-s − 0.702i·11-s + (−0.467 − 0.339i)12-s + (−1.00 − 1.00i)13-s + (0.0824 + 1.04i)14-s + (−0.0434 + 0.575i)15-s + (−0.951 + 0.309i)16-s + (0.552 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.454 + 0.890i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ -0.454 + 0.890i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.383495 - 0.626135i\)
\(L(\frac12)\) \(\approx\) \(0.383495 - 0.626135i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.15 + 1.83i)T \)
3 \( 1 + (2.12 - 2.12i)T \)
5 \( 1 + (-8.47 + 7.28i)T \)
good7 \( 1 + (13.7 + 13.7i)T + 343iT^{2} \)
11 \( 1 + 25.6iT - 1.33e3T^{2} \)
13 \( 1 + (46.9 + 46.9i)T + 2.19e3iT^{2} \)
17 \( 1 + (-38.6 + 38.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 59.7T + 6.85e3T^{2} \)
23 \( 1 + (-32.6 + 32.6i)T - 1.21e4iT^{2} \)
29 \( 1 + 28.5iT - 2.43e4T^{2} \)
31 \( 1 - 274. iT - 2.97e4T^{2} \)
37 \( 1 + (291. - 291. i)T - 5.06e4iT^{2} \)
41 \( 1 - 10.0T + 6.89e4T^{2} \)
43 \( 1 + (-172. + 172. i)T - 7.95e4iT^{2} \)
47 \( 1 + (430. + 430. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-420. - 420. i)T + 1.48e5iT^{2} \)
59 \( 1 - 149.T + 2.05e5T^{2} \)
61 \( 1 - 394.T + 2.26e5T^{2} \)
67 \( 1 + (-428. - 428. i)T + 3.00e5iT^{2} \)
71 \( 1 + 560. iT - 3.57e5T^{2} \)
73 \( 1 + (-480. - 480. i)T + 3.89e5iT^{2} \)
79 \( 1 + 347.T + 4.93e5T^{2} \)
83 \( 1 + (217. - 217. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.44e3iT - 7.04e5T^{2} \)
97 \( 1 + (62.1 - 62.1i)T - 9.12e5iT^{2} \)
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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

−13.90519683086097443905872526853, −12.87139338249471497167503823052, −11.89671492644790558355991730677, −10.32932929166533839832256572468, −9.915637326907432497428913304065, −8.617023675368341925732335454563, −7.00844254466467431850029794160, −5.19261877354895228005518021861, −3.21460637432965774083612575950, −0.69015794705615062217275646607, 2.13426397929536773926973282026, 5.41998989609160287623635033926, 6.50961823997806525036476634739, 7.46437202903299187933758285707, 9.331528161519488295910159373069, 9.949170009997639182491247060302, 11.40247752072768794863799348846, 12.71026113043670640050098753599, 14.13767365612182171814079148736, 14.99079799793137487030146261367

Graph of the $Z$-function along the critical line