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Properties

Label 2-60-20.19-c4-0-9
Degree $2$
Conductor $60$
Sign $0.0573 + 0.998i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.90 + 0.854i)2-s − 5.19·3-s + (14.5 − 6.67i)4-s + (−9.11 + 23.2i)5-s + (20.3 − 4.44i)6-s − 10.5·7-s + (−51.1 + 38.5i)8-s + 27·9-s + (15.7 − 98.7i)10-s − 38.4i·11-s + (−75.5 + 34.7i)12-s − 273. i·13-s + (41.1 − 8.99i)14-s + (47.3 − 120. i)15-s + (166. − 194. i)16-s − 256. i·17-s + ⋯
L(s)  = 1  + (−0.976 + 0.213i)2-s − 0.577·3-s + (0.908 − 0.417i)4-s + (−0.364 + 0.931i)5-s + (0.564 − 0.123i)6-s − 0.214·7-s + (−0.798 + 0.601i)8-s + 0.333·9-s + (0.157 − 0.987i)10-s − 0.317i·11-s + (−0.524 + 0.241i)12-s − 1.61i·13-s + (0.209 − 0.0458i)14-s + (0.210 − 0.537i)15-s + (0.651 − 0.758i)16-s − 0.887i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0573 + 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0573 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0573 + 0.998i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.0573 + 0.998i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.318422 - 0.300659i\)
\(L(\frac12)\) \(\approx\) \(0.318422 - 0.300659i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.90 - 0.854i)T \)
3 \( 1 + 5.19T \)
5 \( 1 + (9.11 - 23.2i)T \)
good7 \( 1 + 10.5T + 2.40e3T^{2} \)
11 \( 1 + 38.4iT - 1.46e4T^{2} \)
13 \( 1 + 273. iT - 2.85e4T^{2} \)
17 \( 1 + 256. iT - 8.35e4T^{2} \)
19 \( 1 + 257. iT - 1.30e5T^{2} \)
23 \( 1 - 168.T + 2.79e5T^{2} \)
29 \( 1 + 684.T + 7.07e5T^{2} \)
31 \( 1 - 754. iT - 9.23e5T^{2} \)
37 \( 1 + 1.65e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.20e3T + 2.82e6T^{2} \)
43 \( 1 - 3.41e3T + 3.41e6T^{2} \)
47 \( 1 + 3.24e3T + 4.87e6T^{2} \)
53 \( 1 - 639. iT - 7.89e6T^{2} \)
59 \( 1 + 6.77e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.88e3T + 1.38e7T^{2} \)
67 \( 1 + 6.75e3T + 2.01e7T^{2} \)
71 \( 1 + 5.87e3iT - 2.54e7T^{2} \)
73 \( 1 - 7.21e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.38e3iT - 3.89e7T^{2} \)
83 \( 1 + 5.47e3T + 4.74e7T^{2} \)
89 \( 1 + 4.60e3T + 6.27e7T^{2} \)
97 \( 1 - 758. iT - 8.85e7T^{2} \)
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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

−14.41573053339398971096166559835, −12.69889116839521678536117243190, −11.28625251458084389404627487402, −10.70690901528056506363592215562, −9.501965046694742587768277373053, −7.88432414864322567410377596120, −6.89545826505964982834335260063, −5.59304194077501891677815791419, −2.95438857526723383685839039805, −0.37037221618933796702926987723, 1.54148620027912411895625343151, 4.17706755467618628476874804533, 6.14198331821583337998672705778, 7.54426507552364746312898649274, 8.861462567619563958741973644201, 9.835913429101238423754567295668, 11.24911747626976155271867729307, 12.07500045835851977518420221834, 13.04148245267852792271383531166, 14.94194791783370527450501868693

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