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Properties

Label 2-60-20.19-c6-0-21
Degree $2$
Conductor $60$
Sign $0.811 + 0.584i$
Analytic cond. $13.8032$
Root an. cond. $3.71527$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.30 + 6.74i)2-s + 15.5·3-s + (−26.8 − 58.0i)4-s + (−108. + 61.3i)5-s + (−67.1 + 105. i)6-s − 118.·7-s + (507. + 69.1i)8-s + 243·9-s + (55.5 − 998. i)10-s + 184. i·11-s + (−418. − 905. i)12-s − 2.86e3i·13-s + (508. − 796. i)14-s + (−1.69e3 + 956. i)15-s + (−2.65e3 + 3.12e3i)16-s − 4.93e3i·17-s + ⋯
L(s)  = 1  + (−0.538 + 0.842i)2-s + 0.577·3-s + (−0.419 − 0.907i)4-s + (−0.871 + 0.490i)5-s + (−0.310 + 0.486i)6-s − 0.344·7-s + (0.990 + 0.135i)8-s + 0.333·9-s + (0.0555 − 0.998i)10-s + 0.138i·11-s + (−0.242 − 0.523i)12-s − 1.30i·13-s + (0.185 − 0.290i)14-s + (−0.502 + 0.283i)15-s + (−0.647 + 0.762i)16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.811 + 0.584i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.917754 - 0.296146i\)
\(L(\frac12)\) \(\approx\) \(0.917754 - 0.296146i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.30 - 6.74i)T \)
3 \( 1 - 15.5T \)
5 \( 1 + (108. - 61.3i)T \)
good7 \( 1 + 118.T + 1.17e5T^{2} \)
11 \( 1 - 184. iT - 1.77e6T^{2} \)
13 \( 1 + 2.86e3iT - 4.82e6T^{2} \)
17 \( 1 + 4.93e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.72e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.95e4T + 1.48e8T^{2} \)
29 \( 1 - 1.24e4T + 5.94e8T^{2} \)
31 \( 1 + 1.58e4iT - 8.87e8T^{2} \)
37 \( 1 + 7.83e3iT - 2.56e9T^{2} \)
41 \( 1 + 1.10e5T + 4.75e9T^{2} \)
43 \( 1 + 3.91e4T + 6.32e9T^{2} \)
47 \( 1 - 1.35e5T + 1.07e10T^{2} \)
53 \( 1 + 2.54e4iT - 2.21e10T^{2} \)
59 \( 1 + 4.06e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.91e5T + 5.15e10T^{2} \)
67 \( 1 + 5.37e4T + 9.04e10T^{2} \)
71 \( 1 - 6.71e4iT - 1.28e11T^{2} \)
73 \( 1 + 3.42e5iT - 1.51e11T^{2} \)
79 \( 1 - 6.70e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.73e5T + 3.26e11T^{2} \)
89 \( 1 + 7.21e5T + 4.96e11T^{2} \)
97 \( 1 + 1.49e6iT - 8.32e11T^{2} \)
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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

−14.03361116598098833104961144343, −12.88195030605632300984139593876, −11.17608233287972933199943688751, −10.01904696541735722032486020413, −8.778763289499363716559265867895, −7.67167933014566834746073571763, −6.77935823230917550160193742163, −4.93247637637340004217508695192, −3.06984436175949190117941114184, −0.48866131903337562642971578389, 1.41814912118645129579021423315, 3.28912722272765389227717736017, 4.45265741719218552203682339580, 7.07760356541105626110260389909, 8.401529863694348035059687978578, 9.120470274757011365298647102035, 10.49228980293821650232046265565, 11.73348568590422504921898541402, 12.62082393396909380432294602423, 13.67548182049861146653254225887

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