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Properties

Label 2-60-20.19-c6-0-4
Degree $2$
Conductor $60$
Sign $0.519 - 0.854i$
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  + (−6.43 + 4.75i)2-s − 15.5·3-s + (18.7 − 61.1i)4-s + (−121. − 30.8i)5-s + (100. − 74.1i)6-s − 549.·7-s + (170. + 482. i)8-s + 243·9-s + (925. − 377. i)10-s − 2.55e3i·11-s + (−291. + 953. i)12-s + 1.09e3i·13-s + (3.53e3 − 2.61e3i)14-s + (1.88e3 + 481. i)15-s + (−3.39e3 − 2.29e3i)16-s + 1.49e3i·17-s + ⋯
L(s)  = 1  + (−0.803 + 0.594i)2-s − 0.577·3-s + (0.292 − 0.956i)4-s + (−0.968 − 0.247i)5-s + (0.464 − 0.343i)6-s − 1.60·7-s + (0.333 + 0.942i)8-s + 0.333·9-s + (0.925 − 0.377i)10-s − 1.91i·11-s + (−0.168 + 0.552i)12-s + 0.498i·13-s + (1.28 − 0.952i)14-s + (0.559 + 0.142i)15-s + (−0.828 − 0.559i)16-s + 0.304i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.519 - 0.854i$
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: \((2,\ 60,\ (\ :3),\ 0.519 - 0.854i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.339784 + 0.190987i\)
\(L(\frac12)\) \(\approx\) \(0.339784 + 0.190987i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.43 - 4.75i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (121. + 30.8i)T \)
good7 \( 1 + 549.T + 1.17e5T^{2} \)
11 \( 1 + 2.55e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.09e3iT - 4.82e6T^{2} \)
17 \( 1 - 1.49e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.51e3iT - 4.70e7T^{2} \)
23 \( 1 - 67.4T + 1.48e8T^{2} \)
29 \( 1 + 6.17e3T + 5.94e8T^{2} \)
31 \( 1 - 2.08e4iT - 8.87e8T^{2} \)
37 \( 1 + 2.66e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.02e5T + 4.75e9T^{2} \)
43 \( 1 + 8.84e3T + 6.32e9T^{2} \)
47 \( 1 + 4.90e4T + 1.07e10T^{2} \)
53 \( 1 + 2.46e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.14e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.51e4T + 5.15e10T^{2} \)
67 \( 1 - 3.80e5T + 9.04e10T^{2} \)
71 \( 1 + 3.23e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.61e5iT - 1.51e11T^{2} \)
79 \( 1 - 7.53e5iT - 2.43e11T^{2} \)
83 \( 1 - 2.36e4T + 3.26e11T^{2} \)
89 \( 1 + 5.59e5T + 4.96e11T^{2} \)
97 \( 1 - 8.43e5iT - 8.32e11T^{2} \)
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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

−14.21275993897365076405294451732, −12.79958959484170905494407334633, −11.53244285839814003929040715395, −10.51381633981650631644302529774, −9.213894241149065912757340591590, −8.089671436724075239955185825375, −6.65419629359998012094932686042, −5.73724717318398578597462186434, −3.58451755413675077557811426444, −0.68029750768281866913358189885, 0.39861077088817332216833165975, 2.77879539748484174981715791136, 4.30258389253974472791842849717, 6.74358941820075386083751969536, 7.48955951961284985264526800471, 9.317756200281953211340559137327, 10.13955136705289526012400703244, 11.33093766580157383184669169473, 12.44354671686960635799374051900, 12.96195754422217663886401376718

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