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Properties

Label 2-60-20.19-c6-0-5
Degree $2$
Conductor $60$
Sign $-0.184 - 0.982i$
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  + (−7.65 − 2.31i)2-s − 15.5·3-s + (53.2 + 35.5i)4-s + (87.3 + 89.3i)5-s + (119. + 36.1i)6-s − 36.3·7-s + (−325. − 395. i)8-s + 243·9-s + (−461. − 886. i)10-s − 995. i·11-s + (−829. − 553. i)12-s − 105. i·13-s + (278. + 84.2i)14-s + (−1.36e3 − 1.39e3i)15-s + (1.57e3 + 3.78e3i)16-s + 1.28e3i·17-s + ⋯
L(s)  = 1  + (−0.957 − 0.289i)2-s − 0.577·3-s + (0.831 + 0.554i)4-s + (0.699 + 0.714i)5-s + (0.552 + 0.167i)6-s − 0.105·7-s + (−0.635 − 0.772i)8-s + 0.333·9-s + (−0.461 − 0.886i)10-s − 0.748i·11-s + (−0.480 − 0.320i)12-s − 0.0482i·13-s + (0.101 + 0.0306i)14-s + (−0.403 − 0.412i)15-s + (0.384 + 0.923i)16-s + 0.261i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.184 - 0.982i$
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: \((2,\ 60,\ (\ :3),\ -0.184 - 0.982i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.449136 + 0.541474i\)
\(L(\frac12)\) \(\approx\) \(0.449136 + 0.541474i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.65 + 2.31i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (-87.3 - 89.3i)T \)
good7 \( 1 + 36.3T + 1.17e5T^{2} \)
11 \( 1 + 995. iT - 1.77e6T^{2} \)
13 \( 1 + 105. iT - 4.82e6T^{2} \)
17 \( 1 - 1.28e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.08e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.33e3T + 1.48e8T^{2} \)
29 \( 1 + 1.42e4T + 5.94e8T^{2} \)
31 \( 1 - 4.05e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.13e4iT - 2.56e9T^{2} \)
41 \( 1 + 6.84e4T + 4.75e9T^{2} \)
43 \( 1 + 6.39e3T + 6.32e9T^{2} \)
47 \( 1 - 5.64e4T + 1.07e10T^{2} \)
53 \( 1 - 6.28e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.32e5iT - 4.21e10T^{2} \)
61 \( 1 + 3.87e5T + 5.15e10T^{2} \)
67 \( 1 - 4.76e5T + 9.04e10T^{2} \)
71 \( 1 - 4.83e5iT - 1.28e11T^{2} \)
73 \( 1 - 6.54e5iT - 1.51e11T^{2} \)
79 \( 1 - 8.01e4iT - 2.43e11T^{2} \)
83 \( 1 + 1.03e6T + 3.26e11T^{2} \)
89 \( 1 - 1.20e6T + 4.96e11T^{2} \)
97 \( 1 + 1.45e6iT - 8.32e11T^{2} \)
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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

−14.16643725829361615796301895645, −12.77583567298049554854573044764, −11.56148769942353378589073190611, −10.57329841440408992210450236989, −9.797077756566510116749078956150, −8.328532419320143063413606397561, −6.85927167760244433689328700863, −5.82659248355695110467291087120, −3.28208160578207558759592104471, −1.52545787736954355678316850610, 0.42923060815660489333953522888, 2.02605908103467967273335356307, 4.94346546785160783824868138298, 6.19810594982442650632396393102, 7.46211659620666072153325465337, 9.033387391584365550178562166290, 9.811484639954253494529204278863, 11.03991619893090468877260214421, 12.23271298930563897486283819177, 13.44753870904937639989326528892

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