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Properties

Label 2-60-20.19-c4-0-22
Degree $2$
Conductor $60$
Sign $-0.896 + 0.443i$
Analytic cond. $6.20219$
Root an. cond. $2.49042$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.44 − 2.03i)2-s − 5.19·3-s + (7.71 − 14.0i)4-s + (−20.5 − 14.2i)5-s + (−17.8 + 10.5i)6-s − 51.0·7-s + (−1.97 − 63.9i)8-s + 27·9-s + (−99.7 − 7.45i)10-s + 37.8i·11-s + (−40.0 + 72.8i)12-s − 201. i·13-s + (−175. + 103. i)14-s + (106. + 74.2i)15-s + (−136. − 216. i)16-s − 370. i·17-s + ⋯
L(s)  = 1  + (0.860 − 0.508i)2-s − 0.577·3-s + (0.482 − 0.876i)4-s + (−0.820 − 0.571i)5-s + (−0.497 + 0.293i)6-s − 1.04·7-s + (−0.0308 − 0.999i)8-s + 0.333·9-s + (−0.997 − 0.0745i)10-s + 0.312i·11-s + (−0.278 + 0.505i)12-s − 1.19i·13-s + (−0.896 + 0.530i)14-s + (0.473 + 0.330i)15-s + (−0.535 − 0.844i)16-s − 1.28i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.896 + 0.443i$
Analytic conductor: \(6.20219\)
Root analytic conductor: \(2.49042\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :2),\ -0.896 + 0.443i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.289678 - 1.23941i\)
\(L(\frac12)\) \(\approx\) \(0.289678 - 1.23941i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.44 + 2.03i)T \)
3 \( 1 + 5.19T \)
5 \( 1 + (20.5 + 14.2i)T \)
good7 \( 1 + 51.0T + 2.40e3T^{2} \)
11 \( 1 - 37.8iT - 1.46e4T^{2} \)
13 \( 1 + 201. iT - 2.85e4T^{2} \)
17 \( 1 + 370. iT - 8.35e4T^{2} \)
19 \( 1 - 474. iT - 1.30e5T^{2} \)
23 \( 1 - 386.T + 2.79e5T^{2} \)
29 \( 1 - 1.16e3T + 7.07e5T^{2} \)
31 \( 1 + 1.45e3iT - 9.23e5T^{2} \)
37 \( 1 + 609. iT - 1.87e6T^{2} \)
41 \( 1 - 731.T + 2.82e6T^{2} \)
43 \( 1 + 1.82e3T + 3.41e6T^{2} \)
47 \( 1 - 1.39e3T + 4.87e6T^{2} \)
53 \( 1 - 3.05e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.23e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.50e3T + 1.38e7T^{2} \)
67 \( 1 + 5.33e3T + 2.01e7T^{2} \)
71 \( 1 + 5.69e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.46e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.14e3iT - 3.89e7T^{2} \)
83 \( 1 - 6.10e3T + 4.74e7T^{2} \)
89 \( 1 - 9.51e3T + 6.27e7T^{2} \)
97 \( 1 + 3.48e3iT - 8.85e7T^{2} \)
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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

−13.49739256887100675447659032617, −12.54562477308700399691648188638, −11.94172290589091226305228253748, −10.63365535087773999472568990157, −9.528819531566792173961114912203, −7.50730148171672499339149925831, −6.02462932345018258177148527256, −4.72227707374227270138657658396, −3.23441321638154317780230618212, −0.58148732453325409207166324206, 3.18450346703584751838609885714, 4.56134273039220826953198104032, 6.39205265977786681507996344607, 6.98443359636029376079464695813, 8.675322588211361760734146886243, 10.60992613073002434406769414371, 11.65292930408173008065337468357, 12.58167604338288808513593481474, 13.68997892744052517329949014537, 14.93935218834111395002287941120

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