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Properties

Label 2-60-20.19-c6-0-18
Degree $2$
Conductor $60$
Sign $0.941 + 0.336i$
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  + (−7.34 − 3.15i)2-s + 15.5·3-s + (44.0 + 46.4i)4-s + (111. − 56.4i)5-s + (−114. − 49.2i)6-s + 231.·7-s + (−176. − 480. i)8-s + 243·9-s + (−998. + 62.8i)10-s + 1.43e3i·11-s + (686. + 723. i)12-s + 1.70e3i·13-s + (−1.70e3 − 731. i)14-s + (1.73e3 − 880. i)15-s + (−217. + 4.09e3i)16-s − 7.56e3i·17-s + ⋯
L(s)  = 1  + (−0.918 − 0.394i)2-s + 0.577·3-s + (0.688 + 0.725i)4-s + (0.892 − 0.451i)5-s + (−0.530 − 0.228i)6-s + 0.675·7-s + (−0.345 − 0.938i)8-s + 0.333·9-s + (−0.998 + 0.0628i)10-s + 1.07i·11-s + (0.397 + 0.418i)12-s + 0.775i·13-s + (−0.620 − 0.266i)14-s + (0.515 − 0.260i)15-s + (−0.0531 + 0.998i)16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.941 + 0.336i$
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: \((2,\ 60,\ (\ :3),\ 0.941 + 0.336i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.83340 - 0.317654i\)
\(L(\frac12)\) \(\approx\) \(1.83340 - 0.317654i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.34 + 3.15i)T \)
3 \( 1 - 15.5T \)
5 \( 1 + (-111. + 56.4i)T \)
good7 \( 1 - 231.T + 1.17e5T^{2} \)
11 \( 1 - 1.43e3iT - 1.77e6T^{2} \)
13 \( 1 - 1.70e3iT - 4.82e6T^{2} \)
17 \( 1 + 7.56e3iT - 2.41e7T^{2} \)
19 \( 1 - 5.56e3iT - 4.70e7T^{2} \)
23 \( 1 - 1.44e4T + 1.48e8T^{2} \)
29 \( 1 - 2.57e4T + 5.94e8T^{2} \)
31 \( 1 + 4.09e4iT - 8.87e8T^{2} \)
37 \( 1 - 6.60e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.06e5T + 4.75e9T^{2} \)
43 \( 1 + 5.83e4T + 6.32e9T^{2} \)
47 \( 1 + 7.50e4T + 1.07e10T^{2} \)
53 \( 1 - 8.18e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.99e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.68e5T + 5.15e10T^{2} \)
67 \( 1 - 3.68e4T + 9.04e10T^{2} \)
71 \( 1 + 4.49e5iT - 1.28e11T^{2} \)
73 \( 1 + 5.85e4iT - 1.51e11T^{2} \)
79 \( 1 - 6.49e5iT - 2.43e11T^{2} \)
83 \( 1 + 4.52e5T + 3.26e11T^{2} \)
89 \( 1 + 2.86e5T + 4.96e11T^{2} \)
97 \( 1 + 3.86e5iT - 8.32e11T^{2} \)
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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

−13.72781900900861177057242409864, −12.53829420321116034816799522829, −11.40833654516361922627408324977, −9.892647305868139290298386922019, −9.299325116750371894753137039219, −8.058019288057582239144952873324, −6.80152450743227104791467838216, −4.68128721681355029583252793994, −2.51128035343146044646503975837, −1.32747050731382066328903216668, 1.26149484208992569283160345613, 2.82547533341346887593125237377, 5.45433775708601546169559762631, 6.73001268699313145804197061304, 8.173879930744330875485024715162, 8.992966645650895570615562898425, 10.39539613307233654803321532272, 11.05564502196987472007334069455, 13.02056913240866737113820425808, 14.27682526798351755818429789182

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