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Properties

Label 2-60-20.19-c6-0-8
Degree $2$
Conductor $60$
Sign $0.699 - 0.715i$
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 + (843. + 537. 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 + (0.843 + 0.537i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.699 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.699 - 0.715i$
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: \((2,\ 60,\ (\ :3),\ 0.699 - 0.715i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.38002 + 0.580751i\)
\(L(\frac12)\) \(\approx\) \(1.38002 + 0.580751i\)
\(L(4)\) 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

−14.08794516435638180726380694839, −12.70955097007008291251965213749, −11.80172347766079676616894792911, −10.45534593400701110071302228961, −9.691774867036197513935802033461, −8.199426863269900622806903300993, −6.71159582373534739507563822432, −4.57915082564281716262020429792, −3.26084732194647021266735011390, −1.90749723994650504163527155930, 0.55053231049705023494936354681, 3.41560343500632545614024461765, 4.84330121096868582080489553685, 6.25205704117392986234006135288, 7.953301150611833170891263559551, 8.549833230956204461882281422579, 9.751912648895054426299760755874, 11.71656361826198339589763945567, 13.05877951143173059945153169432, 13.67698410836826814385029490005

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