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Properties

Label 2-60-4.3-c4-0-7
Degree $2$
Conductor $60$
Sign $0.892 + 0.452i$
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  + (−3.40 + 2.09i)2-s + 5.19i·3-s + (7.23 − 14.2i)4-s − 11.1·5-s + (−10.8 − 17.7i)6-s − 61.3i·7-s + (5.23 + 63.7i)8-s − 27·9-s + (38.1 − 23.4i)10-s − 74.1i·11-s + (74.1 + 37.5i)12-s + 181.·13-s + (128. + 209. i)14-s − 58.0i·15-s + (−151. − 206. i)16-s + 516.·17-s + ⋯
L(s)  = 1  + (−0.852 + 0.523i)2-s + 0.577i·3-s + (0.452 − 0.892i)4-s − 0.447·5-s + (−0.302 − 0.491i)6-s − 1.25i·7-s + (0.0817 + 0.996i)8-s − 0.333·9-s + (0.381 − 0.234i)10-s − 0.612i·11-s + (0.515 + 0.260i)12-s + 1.07·13-s + (0.655 + 1.06i)14-s − 0.258i·15-s + (−0.591 − 0.806i)16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.892 + 0.452i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.892 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.892 + 0.452i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ 0.892 + 0.452i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.872274 - 0.208396i\)
\(L(\frac12)\) \(\approx\) \(0.872274 - 0.208396i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.40 - 2.09i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 + 11.1T \)
good7 \( 1 + 61.3iT - 2.40e3T^{2} \)
11 \( 1 + 74.1iT - 1.46e4T^{2} \)
13 \( 1 - 181.T + 2.85e4T^{2} \)
17 \( 1 - 516.T + 8.35e4T^{2} \)
19 \( 1 + 407. iT - 1.30e5T^{2} \)
23 \( 1 - 7.48iT - 2.79e5T^{2} \)
29 \( 1 + 1.47e3T + 7.07e5T^{2} \)
31 \( 1 + 1.04e3iT - 9.23e5T^{2} \)
37 \( 1 + 667.T + 1.87e6T^{2} \)
41 \( 1 - 1.21e3T + 2.82e6T^{2} \)
43 \( 1 + 987. iT - 3.41e6T^{2} \)
47 \( 1 + 2.94e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.28e3T + 7.89e6T^{2} \)
59 \( 1 - 390. iT - 1.21e7T^{2} \)
61 \( 1 - 4.10e3T + 1.38e7T^{2} \)
67 \( 1 - 6.16e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.46e3iT - 2.54e7T^{2} \)
73 \( 1 - 3.36e3T + 2.83e7T^{2} \)
79 \( 1 - 7.06e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.97e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.23e4T + 6.27e7T^{2} \)
97 \( 1 - 2.51e3T + 8.85e7T^{2} \)
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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

−14.50372164544357557927877766362, −13.46822274915939694281631998076, −11.40728884909490095450938546235, −10.68713329017274559237608460846, −9.553605319282176384421977237148, −8.252960788694199965849058505174, −7.18311355630751886201646062662, −5.59694487109694623621935817404, −3.73282994970699539423133902593, −0.73105051924734933235887898313, 1.57850078864724069196999758801, 3.35350334573562469671628199361, 5.89122247215550151504611793967, 7.54422399091393815058090018528, 8.470702651500622698232466413939, 9.656777192950138704658321456788, 11.10846105797135162897706819820, 12.19390678575729951874126633787, 12.68848882235262082293459409399, 14.48371844292384799641850561567

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