Properties

Label 2-60-4.3-c6-0-11
Degree $2$
Conductor $60$
Sign $0.930 + 0.365i$
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  + (−6.61 + 4.50i)2-s − 15.5i·3-s + (23.4 − 59.5i)4-s − 55.9·5-s + (70.2 + 103. i)6-s + 489. i·7-s + (113. + 499. i)8-s − 243·9-s + (369. − 251. i)10-s − 1.67e3i·11-s + (−928. − 364. i)12-s − 528.·13-s + (−2.20e3 − 3.23e3i)14-s + 871. i·15-s + (−2.99e3 − 2.78e3i)16-s + 8.23e3·17-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s − 0.577i·3-s + (0.365 − 0.930i)4-s − 0.447·5-s + (0.325 + 0.477i)6-s + 1.42i·7-s + (0.221 + 0.975i)8-s − 0.333·9-s + (0.369 − 0.251i)10-s − 1.25i·11-s + (−0.537 − 0.211i)12-s − 0.240·13-s + (−0.803 − 1.17i)14-s + 0.258i·15-s + (−0.732 − 0.680i)16-s + 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.930 + 0.365i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.930 + 0.365i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.01248 - 0.191840i\)
\(L(\frac12)\) \(\approx\) \(1.01248 - 0.191840i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.61 - 4.50i)T \)
3 \( 1 + 15.5iT \)
5 \( 1 + 55.9T \)
good7 \( 1 - 489. iT - 1.17e5T^{2} \)
11 \( 1 + 1.67e3iT - 1.77e6T^{2} \)
13 \( 1 + 528.T + 4.82e6T^{2} \)
17 \( 1 - 8.23e3T + 2.41e7T^{2} \)
19 \( 1 - 3.89e3iT - 4.70e7T^{2} \)
23 \( 1 + 2.31e4iT - 1.48e8T^{2} \)
29 \( 1 - 4.39e4T + 5.94e8T^{2} \)
31 \( 1 + 1.82e4iT - 8.87e8T^{2} \)
37 \( 1 - 7.60e4T + 2.56e9T^{2} \)
41 \( 1 + 7.73e4T + 4.75e9T^{2} \)
43 \( 1 + 4.50e4iT - 6.32e9T^{2} \)
47 \( 1 + 7.96e4iT - 1.07e10T^{2} \)
53 \( 1 - 4.61e4T + 2.21e10T^{2} \)
59 \( 1 + 1.22e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.73e5T + 5.15e10T^{2} \)
67 \( 1 - 3.54e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.73e4iT - 1.28e11T^{2} \)
73 \( 1 - 3.66e5T + 1.51e11T^{2} \)
79 \( 1 + 1.10e5iT - 2.43e11T^{2} \)
83 \( 1 - 3.53e5iT - 3.26e11T^{2} \)
89 \( 1 + 9.65e5T + 4.96e11T^{2} \)
97 \( 1 + 3.68e5T + 8.32e11T^{2} \)
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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

−14.14587653668797199496699293087, −12.38310866443653543503230180935, −11.57275274086312506295714298799, −10.12829696976979736450092320147, −8.601118616662853942768724263836, −8.096883659932260041332595928921, −6.44254558144719309935567447496, −5.46241236680642778835552788666, −2.65639111671225401644167037837, −0.72911237579301055834710254678, 1.07563813034733095559397262186, 3.30095267544094975908545643733, 4.54680114619142833054973098739, 7.11281710929453852689060864497, 7.927809397820451453100878775834, 9.667285116877294581335186007015, 10.21483870950444086722007932306, 11.41338771885775972723435729100, 12.46869665185213792536920709832, 13.85603992670610357061535718156

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