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Properties

Label 2-60-4.3-c2-0-1
Degree $2$
Conductor $60$
Sign $-0.427 - 0.904i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.438 + 1.95i)2-s + 1.73i·3-s + (−3.61 + 1.71i)4-s + 2.23·5-s + (−3.37 + 0.758i)6-s + 6.33i·7-s + (−4.92 − 6.30i)8-s − 2.99·9-s + (0.979 + 4.36i)10-s − 9.27i·11-s + (−2.96 − 6.26i)12-s + 18.5·13-s + (−12.3 + 2.77i)14-s + 3.87i·15-s + (10.1 − 12.3i)16-s + 13.9·17-s + ⋯
L(s)  = 1  + (0.219 + 0.975i)2-s + 0.577i·3-s + (−0.904 + 0.427i)4-s + 0.447·5-s + (−0.563 + 0.126i)6-s + 0.904i·7-s + (−0.615 − 0.788i)8-s − 0.333·9-s + (0.0979 + 0.436i)10-s − 0.843i·11-s + (−0.246 − 0.521i)12-s + 1.42·13-s + (−0.882 + 0.198i)14-s + 0.258i·15-s + (0.634 − 0.772i)16-s + 0.818·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.427 - 0.904i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.681091 + 1.07552i\)
\(L(\frac12)\) \(\approx\) \(0.681091 + 1.07552i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.438 - 1.95i)T \)
3 \( 1 - 1.73iT \)
5 \( 1 - 2.23T \)
good7 \( 1 - 6.33iT - 49T^{2} \)
11 \( 1 + 9.27iT - 121T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 - 13.9T + 289T^{2} \)
19 \( 1 - 17.2iT - 361T^{2} \)
23 \( 1 + 33.7iT - 529T^{2} \)
29 \( 1 + 28.6T + 841T^{2} \)
31 \( 1 + 23.4iT - 961T^{2} \)
37 \( 1 + 67.3T + 1.36e3T^{2} \)
41 \( 1 + 44.0T + 1.68e3T^{2} \)
43 \( 1 - 50.2iT - 1.84e3T^{2} \)
47 \( 1 + 31.1iT - 2.20e3T^{2} \)
53 \( 1 - 81.6T + 2.80e3T^{2} \)
59 \( 1 + 19.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.1T + 3.72e3T^{2} \)
67 \( 1 + 4.49iT - 4.48e3T^{2} \)
71 \( 1 + 13.3iT - 5.04e3T^{2} \)
73 \( 1 - 40.8T + 5.32e3T^{2} \)
79 \( 1 + 141. iT - 6.24e3T^{2} \)
83 \( 1 - 69.8iT - 6.88e3T^{2} \)
89 \( 1 + 46.3T + 7.92e3T^{2} \)
97 \( 1 - 68.5T + 9.40e3T^{2} \)
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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

−15.23852027462976070997756152882, −14.30090505194040021162333962655, −13.30449264297834966319924241178, −11.99993788810681609870392801529, −10.41809610707503167198972507060, −9.020735408701690639123978765274, −8.265472651007401278552177569076, −6.24057895797071930214489246274, −5.44038355124807949876542633544, −3.57812825554917118722437595741, 1.48753735575918638957670754400, 3.62191835015240483968065384459, 5.44242165832381185361714112609, 7.15178378857471529654629147460, 8.805488673974056324019446471669, 10.09297215465649447814753484238, 11.10268012491640816014717248828, 12.29527908748590497283438806688, 13.47872910896374049771341996243, 13.84348089718287515247501491800

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