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Properties

Label 2-60-15.14-c6-0-7
Degree $2$
Conductor $60$
Sign $0.830 + 0.556i$
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  + (−7.66 + 25.8i)3-s + (37.2 − 119. i)5-s + 46.7i·7-s + (−611. − 396. i)9-s − 448. i·11-s − 2.07e3i·13-s + (2.80e3 + 1.87e3i)15-s + 5.98e3·17-s + 7.40e3·19-s + (−1.20e3 − 358. i)21-s + 1.71e4·23-s + (−1.28e4 − 8.89e3i)25-s + (1.49e4 − 1.27e4i)27-s − 3.75e4i·29-s − 1.79e4·31-s + ⋯
L(s)  = 1  + (−0.283 + 0.958i)3-s + (0.298 − 0.954i)5-s + 0.136i·7-s + (−0.838 − 0.544i)9-s − 0.337i·11-s − 0.943i·13-s + (0.830 + 0.556i)15-s + 1.21·17-s + 1.07·19-s + (−0.130 − 0.0386i)21-s + 1.41·23-s + (−0.822 − 0.569i)25-s + (0.759 − 0.649i)27-s − 1.53i·29-s − 0.603·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(13.8032\)
Root analytic conductor: \(3.71527\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3),\ 0.830 + 0.556i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.51538 - 0.460884i\)
\(L(\frac12)\) \(\approx\) \(1.51538 - 0.460884i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (7.66 - 25.8i)T \)
5 \( 1 + (-37.2 + 119. i)T \)
good7 \( 1 - 46.7iT - 1.17e5T^{2} \)
11 \( 1 + 448. iT - 1.77e6T^{2} \)
13 \( 1 + 2.07e3iT - 4.82e6T^{2} \)
17 \( 1 - 5.98e3T + 2.41e7T^{2} \)
19 \( 1 - 7.40e3T + 4.70e7T^{2} \)
23 \( 1 - 1.71e4T + 1.48e8T^{2} \)
29 \( 1 + 3.75e4iT - 5.94e8T^{2} \)
31 \( 1 + 1.79e4T + 8.87e8T^{2} \)
37 \( 1 - 3.62e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.28e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.50e4iT - 6.32e9T^{2} \)
47 \( 1 + 5.18e4T + 1.07e10T^{2} \)
53 \( 1 + 2.70e5T + 2.21e10T^{2} \)
59 \( 1 + 2.18e5iT - 4.21e10T^{2} \)
61 \( 1 - 3.83e4T + 5.15e10T^{2} \)
67 \( 1 - 2.59e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.52e5iT - 1.28e11T^{2} \)
73 \( 1 + 7.84e4iT - 1.51e11T^{2} \)
79 \( 1 + 4.10e5T + 2.43e11T^{2} \)
83 \( 1 - 6.30e5T + 3.26e11T^{2} \)
89 \( 1 + 1.45e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.76e6iT - 8.32e11T^{2} \)
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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

−13.76381974733105563114374206995, −12.52768797256542307750600724911, −11.44026463690614740310653337825, −10.10720524853527832749833852067, −9.236597728176942654829807548428, −7.992113768931619523468389994083, −5.80663923062002037365239303184, −4.96757489015422411929695298616, −3.28547707739561311168138715923, −0.76217865401921953689180344574, 1.44088830878794667235636984045, 3.08137831427016832106674978775, 5.40592153581021189782068961201, 6.80005207674223160918129403326, 7.53330330660325871348191153935, 9.293117296756815684722835263189, 10.72131724529907399451288620574, 11.69482990115730359298576394773, 12.82880042903239322963331050197, 14.04105619988588864108759215407

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