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Properties

Label 2-60-20.19-c6-0-14
Degree $2$
Conductor $60$
Sign $0.0511 + 0.998i$
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  + (−1.63 − 7.83i)2-s − 15.5·3-s + (−58.6 + 25.6i)4-s + (−44.2 + 116. i)5-s + (25.5 + 122. i)6-s + 42.6·7-s + (297. + 417. i)8-s + 243·9-s + (987. + 154. i)10-s − 1.84e3i·11-s + (913. − 400. i)12-s + 887. i·13-s + (−69.9 − 334. i)14-s + (689. − 1.82e3i)15-s + (2.77e3 − 3.00e3i)16-s + 798. i·17-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)2-s − 0.577·3-s + (−0.916 + 0.401i)4-s + (−0.353 + 0.935i)5-s + (0.118 + 0.565i)6-s + 0.124·7-s + (0.580 + 0.814i)8-s + 0.333·9-s + (0.987 + 0.154i)10-s − 1.38i·11-s + (0.528 − 0.231i)12-s + 0.403i·13-s + (−0.0254 − 0.121i)14-s + (0.204 − 0.540i)15-s + (0.678 − 0.734i)16-s + 0.162i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0511 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0511 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.0511 + 0.998i$
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: \((2,\ 60,\ (\ :3),\ 0.0511 + 0.998i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.719096 - 0.683206i\)
\(L(\frac12)\) \(\approx\) \(0.719096 - 0.683206i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.63 + 7.83i)T \)
3 \( 1 + 15.5T \)
5 \( 1 + (44.2 - 116. i)T \)
good7 \( 1 - 42.6T + 1.17e5T^{2} \)
11 \( 1 + 1.84e3iT - 1.77e6T^{2} \)
13 \( 1 - 887. iT - 4.82e6T^{2} \)
17 \( 1 - 798. iT - 2.41e7T^{2} \)
19 \( 1 + 2.30e3iT - 4.70e7T^{2} \)
23 \( 1 - 7.16e3T + 1.48e8T^{2} \)
29 \( 1 - 3.71e4T + 5.94e8T^{2} \)
31 \( 1 + 4.76e4iT - 8.87e8T^{2} \)
37 \( 1 + 3.06e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.12e4T + 4.75e9T^{2} \)
43 \( 1 - 1.35e5T + 6.32e9T^{2} \)
47 \( 1 - 1.57e5T + 1.07e10T^{2} \)
53 \( 1 - 1.40e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.50e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.97e4T + 5.15e10T^{2} \)
67 \( 1 + 1.88e5T + 9.04e10T^{2} \)
71 \( 1 + 3.00e5iT - 1.28e11T^{2} \)
73 \( 1 + 1.37e5iT - 1.51e11T^{2} \)
79 \( 1 + 3.92e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.61e5T + 3.26e11T^{2} \)
89 \( 1 - 9.44e5T + 4.96e11T^{2} \)
97 \( 1 + 1.18e6iT - 8.32e11T^{2} \)
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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

−13.46052914819120974177404641888, −12.07054539613580987960454957459, −11.16098814290862635383482065304, −10.57021866998281310357973383563, −9.075449127962604599714032079845, −7.66265092418681494240278185807, −6.00688668070715772278770330034, −4.18703880319791321270685976379, −2.73956305443904731608678301457, −0.64206327988406339449509102547, 1.03769825498458828403661295030, 4.41039991829182178747578069109, 5.30036321847608350788004493329, 6.86219537949523915892499041169, 8.012964216558959056992539594607, 9.261964007842407661354386308589, 10.42220991797009770192944401973, 12.18043386841212832490148225792, 12.90225974704113481128586509274, 14.33539546393516530224809570305

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