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Properties

Label 2-60-60.23-c2-0-12
Degree $2$
Conductor $60$
Sign $0.996 + 0.0890i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.81 + 0.837i)2-s + (−1.32 − 2.69i)3-s + (2.59 + 3.04i)4-s + (3.21 − 3.82i)5-s + (−0.144 − 5.99i)6-s + (3.54 + 3.54i)7-s + (2.16 + 7.70i)8-s + (−5.50 + 7.12i)9-s + (9.04 − 4.26i)10-s − 16.8·11-s + (4.76 − 11.0i)12-s + (−8.64 − 8.64i)13-s + (3.46 + 9.40i)14-s + (−14.5 − 3.59i)15-s + (−2.51 + 15.8i)16-s + (9.72 + 9.72i)17-s + ⋯
L(s)  = 1  + (0.908 + 0.418i)2-s + (−0.440 − 0.897i)3-s + (0.649 + 0.760i)4-s + (0.642 − 0.765i)5-s + (−0.0241 − 0.999i)6-s + (0.506 + 0.506i)7-s + (0.270 + 0.962i)8-s + (−0.611 + 0.791i)9-s + (0.904 − 0.426i)10-s − 1.53·11-s + (0.396 − 0.917i)12-s + (−0.665 − 0.665i)13-s + (0.247 + 0.671i)14-s + (−0.970 − 0.239i)15-s + (−0.157 + 0.987i)16-s + (0.572 + 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0890i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.996 + 0.0890i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ 0.996 + 0.0890i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.75401 - 0.0782535i\)
\(L(\frac12)\) \(\approx\) \(1.75401 - 0.0782535i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.81 - 0.837i)T \)
3 \( 1 + (1.32 + 2.69i)T \)
5 \( 1 + (-3.21 + 3.82i)T \)
good7 \( 1 + (-3.54 - 3.54i)T + 49iT^{2} \)
11 \( 1 + 16.8T + 121T^{2} \)
13 \( 1 + (8.64 + 8.64i)T + 169iT^{2} \)
17 \( 1 + (-9.72 - 9.72i)T + 289iT^{2} \)
19 \( 1 + 4.78T + 361T^{2} \)
23 \( 1 + (-13.5 - 13.5i)T + 529iT^{2} \)
29 \( 1 + 14.8T + 841T^{2} \)
31 \( 1 + 14.0iT - 961T^{2} \)
37 \( 1 + (10.1 - 10.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 6.08iT - 1.68e3T^{2} \)
43 \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-17.6 + 17.6i)T - 2.20e3iT^{2} \)
53 \( 1 + (-16.2 + 16.2i)T - 2.80e3iT^{2} \)
59 \( 1 + 4.37iT - 3.48e3T^{2} \)
61 \( 1 - 8.52T + 3.72e3T^{2} \)
67 \( 1 + (-53.9 - 53.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 36.6T + 5.04e3T^{2} \)
73 \( 1 + (12.6 + 12.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 88.4T + 6.24e3T^{2} \)
83 \( 1 + (63.7 + 63.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 115.T + 7.92e3T^{2} \)
97 \( 1 + (85.3 - 85.3i)T - 9.40e3iT^{2} \)
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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

−14.69589651576792023551328624062, −13.38969430857016189816585164307, −12.83591965104320046268879164851, −11.95125582432890918280441327873, −10.55224570876607560603361048422, −8.406244981614797845874631846214, −7.47171797878368354081697338779, −5.69180272539289761269231536348, −5.18138881738416578542103710262, −2.32250251657023191092353524207, 2.79986328935667326322719000940, 4.61354591139269408089694604312, 5.69451462441382793769758550048, 7.22133624888605521951590816149, 9.635335358423900827118720768920, 10.57572129805313411094374143342, 11.19443283792595130915174497878, 12.62545179136878593033008985829, 13.98727648185475614567469209409, 14.64699260505563166461940876961

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