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Properties

Label 2-60-20.19-c4-0-1
Degree $2$
Conductor $60$
Sign $-0.942 + 0.333i$
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  + (0.828 + 3.91i)2-s + 5.19·3-s + (−14.6 + 6.48i)4-s + (−24.9 − 1.93i)5-s + (4.30 + 20.3i)6-s − 67.1·7-s + (−37.4 − 51.8i)8-s + 27·9-s + (−13.0 − 99.1i)10-s + 173. i·11-s + (−76.0 + 33.6i)12-s + 157. i·13-s + (−55.6 − 262. i)14-s + (−129. − 10.0i)15-s + (171. − 189. i)16-s − 469. i·17-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + 0.577·3-s + (−0.914 + 0.405i)4-s + (−0.997 − 0.0772i)5-s + (0.119 + 0.564i)6-s − 1.37·7-s + (−0.585 − 0.810i)8-s + 0.333·9-s + (−0.130 − 0.991i)10-s + 1.43i·11-s + (−0.527 + 0.234i)12-s + 0.929i·13-s + (−0.283 − 1.34i)14-s + (−0.575 − 0.0445i)15-s + (0.671 − 0.741i)16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.942 + 0.333i$
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: \((2,\ 60,\ (\ :2),\ -0.942 + 0.333i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.119490 - 0.696069i\)
\(L(\frac12)\) \(\approx\) \(0.119490 - 0.696069i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.828 - 3.91i)T \)
3 \( 1 - 5.19T \)
5 \( 1 + (24.9 + 1.93i)T \)
good7 \( 1 + 67.1T + 2.40e3T^{2} \)
11 \( 1 - 173. iT - 1.46e4T^{2} \)
13 \( 1 - 157. iT - 2.85e4T^{2} \)
17 \( 1 + 469. iT - 8.35e4T^{2} \)
19 \( 1 - 286. iT - 1.30e5T^{2} \)
23 \( 1 - 41.9T + 2.79e5T^{2} \)
29 \( 1 + 817.T + 7.07e5T^{2} \)
31 \( 1 - 1.71e3iT - 9.23e5T^{2} \)
37 \( 1 - 76.9iT - 1.87e6T^{2} \)
41 \( 1 + 192.T + 2.82e6T^{2} \)
43 \( 1 - 1.07e3T + 3.41e6T^{2} \)
47 \( 1 + 2.69e3T + 4.87e6T^{2} \)
53 \( 1 + 442. iT - 7.89e6T^{2} \)
59 \( 1 - 573. iT - 1.21e7T^{2} \)
61 \( 1 - 68.2T + 1.38e7T^{2} \)
67 \( 1 + 5.28e3T + 2.01e7T^{2} \)
71 \( 1 + 5.77e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.12e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.12e3iT - 3.89e7T^{2} \)
83 \( 1 - 4.32e3T + 4.74e7T^{2} \)
89 \( 1 + 6.04e3T + 6.27e7T^{2} \)
97 \( 1 - 2.94e3iT - 8.85e7T^{2} \)
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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

−15.06133358819336107721338554316, −14.06425319619841919732058696100, −12.83265736289159337040295334840, −12.05194600425946825922010664620, −9.817040720736929943537015485454, −9.002955233992528113674591557825, −7.45990858035213947858158030913, −6.79153362380632938615911495095, −4.69109271946639587988483638004, −3.38706013371632810463249593288, 0.34539145664032141927058534695, 3.03560346394844785639814925393, 3.85199885058796144376184201759, 6.01808377924332481494902104708, 8.021836751027541211023499919681, 9.074680883919376477425953911719, 10.39258672367249418888715280182, 11.38845956829662190248092639144, 12.84614542041481768192393657780, 13.23979550277047720175458220014

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