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Properties

Label 2-60-60.59-c5-0-18
Degree $2$
Conductor $60$
Sign $0.996 + 0.0795i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.08 + 4.73i)2-s + (−14.4 + 5.82i)3-s + (−12.9 − 29.2i)4-s + (−43.0 + 35.7i)5-s + (17.0 − 86.5i)6-s − 168.·7-s + (178. + 29.2i)8-s + (175. − 168. i)9-s + (−36.3 − 314. i)10-s − 165.·11-s + (357. + 348. i)12-s + 182. i·13-s + (519. − 796. i)14-s + (413. − 766. i)15-s + (−690. + 756. i)16-s + 1.80e3·17-s + ⋯
L(s)  = 1  + (−0.546 + 0.837i)2-s + (−0.927 + 0.373i)3-s + (−0.403 − 0.914i)4-s + (−0.769 + 0.638i)5-s + (0.193 − 0.981i)6-s − 1.29·7-s + (0.986 + 0.161i)8-s + (0.720 − 0.693i)9-s + (−0.114 − 0.993i)10-s − 0.412·11-s + (0.716 + 0.697i)12-s + 0.299i·13-s + (0.708 − 1.08i)14-s + (0.474 − 0.879i)15-s + (−0.674 + 0.738i)16-s + 1.51·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.996 + 0.0795i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ 0.996 + 0.0795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.344956 - 0.0137410i\)
\(L(\frac12)\) \(\approx\) \(0.344956 - 0.0137410i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.08 - 4.73i)T \)
3 \( 1 + (14.4 - 5.82i)T \)
5 \( 1 + (43.0 - 35.7i)T \)
good7 \( 1 + 168.T + 1.68e4T^{2} \)
11 \( 1 + 165.T + 1.61e5T^{2} \)
13 \( 1 - 182. iT - 3.71e5T^{2} \)
17 \( 1 - 1.80e3T + 1.41e6T^{2} \)
19 \( 1 - 1.17e3iT - 2.47e6T^{2} \)
23 \( 1 - 391. iT - 6.43e6T^{2} \)
29 \( 1 + 382. iT - 2.05e7T^{2} \)
31 \( 1 + 8.73e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.38e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.35e4iT - 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 + 1.60e4iT - 2.29e8T^{2} \)
53 \( 1 - 8.22e3T + 4.18e8T^{2} \)
59 \( 1 - 7.47e3T + 7.14e8T^{2} \)
61 \( 1 + 1.23e4T + 8.44e8T^{2} \)
67 \( 1 - 6.06e4T + 1.35e9T^{2} \)
71 \( 1 - 2.83e4T + 1.80e9T^{2} \)
73 \( 1 - 4.02e4iT - 2.07e9T^{2} \)
79 \( 1 + 2.46e4iT - 3.07e9T^{2} \)
83 \( 1 - 2.54e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.57e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.34e5iT - 8.58e9T^{2} \)
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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

−14.52355405088394973968626797491, −12.90439738017621016197319575819, −11.61046914169026930516812405477, −10.31170400473931038364291686855, −9.646233669391306197436034171362, −7.79609701899328070358974285194, −6.67301024214274581131037536997, −5.63513084418788209539009608791, −3.79764727638222416862038107792, −0.31466497753720242375392282750, 0.909456805106745314620528734920, 3.31549737844934376359287499918, 5.06345471849706489840172670900, 6.96358810764061409536733504663, 8.201712384773643777278730333529, 9.698933711467595255714556669034, 10.70178604248579154841863539437, 12.01651208369247507649100866980, 12.54439557981629250991169744147, 13.44011534037898561237961515110

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