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Properties

Label 2-60-4.3-c6-0-8
Degree $2$
Conductor $60$
Sign $0.336 - 0.941i$
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  + (−1.36 + 7.88i)2-s − 15.5i·3-s + (−60.2 − 21.5i)4-s − 55.9·5-s + (122. + 21.2i)6-s − 86.8i·7-s + (251. − 445. i)8-s − 243·9-s + (76.2 − 440. i)10-s + 1.54e3i·11-s + (−335. + 939. i)12-s + 3.18e3·13-s + (685. + 118. i)14-s + 871. i·15-s + (3.17e3 + 2.59e3i)16-s + 4.06e3·17-s + ⋯
L(s)  = 1  + (−0.170 + 0.985i)2-s − 0.577i·3-s + (−0.941 − 0.336i)4-s − 0.447·5-s + (0.568 + 0.0984i)6-s − 0.253i·7-s + (0.491 − 0.870i)8-s − 0.333·9-s + (0.0762 − 0.440i)10-s + 1.16i·11-s + (−0.193 + 0.543i)12-s + 1.45·13-s + (0.249 + 0.0431i)14-s + 0.258i·15-s + (0.774 + 0.632i)16-s + 0.826·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.336 - 0.941i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.10772 + 0.780926i\)
\(L(\frac12)\) \(\approx\) \(1.10772 + 0.780926i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 7.88i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 + 86.8iT - 1.17e5T^{2} \)
11 \( 1 - 1.54e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.18e3T + 4.82e6T^{2} \)
17 \( 1 - 4.06e3T + 2.41e7T^{2} \)
19 \( 1 + 536. iT - 4.70e7T^{2} \)
23 \( 1 - 1.81e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.49e4T + 5.94e8T^{2} \)
31 \( 1 - 1.17e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.49e4T + 2.56e9T^{2} \)
41 \( 1 - 1.08e5T + 4.75e9T^{2} \)
43 \( 1 + 9.19e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.72e4iT - 1.07e10T^{2} \)
53 \( 1 - 2.38e5T + 2.21e10T^{2} \)
59 \( 1 + 9.56e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.44e5T + 5.15e10T^{2} \)
67 \( 1 - 4.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.63e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.25e5T + 1.51e11T^{2} \)
79 \( 1 + 3.18e5iT - 2.43e11T^{2} \)
83 \( 1 - 8.80e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.98e5T + 4.96e11T^{2} \)
97 \( 1 + 1.41e5T + 8.32e11T^{2} \)
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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

−14.11600479360419159716195677707, −13.19969041126830326724374113886, −12.02707739454258926463540812832, −10.44984298681541204244598686065, −9.040939955616349645684476725181, −7.82389581149830406699046433375, −6.95469562244644218583039381524, −5.56276520201170988893439081887, −3.89145772640415722824630407462, −1.14614307652854995417540020510, 0.789192531248572452227288901324, 3.01507353620983134478029322297, 4.16490883656515220952025054133, 5.83192225355967234260097673711, 8.198100163716335991327200220518, 8.949052802710704263790584889045, 10.43661543893203528973539564289, 11.18188385855204778461689970456, 12.24682211113140152438383347154, 13.52902027591685293647824703563

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