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Properties

Label 2-24-8.3-c10-0-1
Degree $2$
Conductor $24$
Sign $-0.190 + 0.981i$
Analytic cond. $15.2485$
Root an. cond. $3.90494$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.04 + 31.9i)2-s − 140.·3-s + (−1.01e3 − 130. i)4-s + 5.06e3i·5-s + (287. − 4.48e3i)6-s + 3.13e4i·7-s + (6.25e3 − 3.21e4i)8-s + 1.96e4·9-s + (−1.61e5 − 1.03e4i)10-s − 1.16e5·11-s + (1.42e5 + 1.83e4i)12-s − 2.17e5i·13-s + (−1.00e6 − 6.41e4i)14-s − 7.10e5i·15-s + (1.01e6 + 2.65e5i)16-s − 8.43e5·17-s + ⋯
L(s)  = 1  + (−0.0640 + 0.997i)2-s − 0.577·3-s + (−0.991 − 0.127i)4-s + 1.61i·5-s + (0.0369 − 0.576i)6-s + 1.86i·7-s + (0.190 − 0.981i)8-s + 0.333·9-s + (−1.61 − 0.103i)10-s − 0.723·11-s + (0.572 + 0.0737i)12-s − 0.584i·13-s + (−1.86 − 0.119i)14-s − 0.935i·15-s + (0.967 + 0.253i)16-s − 0.594·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(15.2485\)
Root analytic conductor: \(3.90494\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :5),\ -0.190 + 0.981i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.460477 - 0.558712i\)
\(L(\frac12)\) \(\approx\) \(0.460477 - 0.558712i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.04 - 31.9i)T \)
3 \( 1 + 140.T \)
good5 \( 1 - 5.06e3iT - 9.76e6T^{2} \)
7 \( 1 - 3.13e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.16e5T + 2.59e10T^{2} \)
13 \( 1 + 2.17e5iT - 1.37e11T^{2} \)
17 \( 1 + 8.43e5T + 2.01e12T^{2} \)
19 \( 1 - 3.50e6T + 6.13e12T^{2} \)
23 \( 1 + 1.01e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.04e7iT - 4.20e14T^{2} \)
31 \( 1 - 2.07e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.54e7iT - 4.80e15T^{2} \)
41 \( 1 - 5.15e7T + 1.34e16T^{2} \)
43 \( 1 - 1.39e8T + 2.16e16T^{2} \)
47 \( 1 - 3.36e8iT - 5.25e16T^{2} \)
53 \( 1 - 1.67e8iT - 1.74e17T^{2} \)
59 \( 1 + 8.34e8T + 5.11e17T^{2} \)
61 \( 1 + 1.51e9iT - 7.13e17T^{2} \)
67 \( 1 + 1.06e9T + 1.82e18T^{2} \)
71 \( 1 - 7.62e8iT - 3.25e18T^{2} \)
73 \( 1 - 3.10e9T + 4.29e18T^{2} \)
79 \( 1 - 4.56e9iT - 9.46e18T^{2} \)
83 \( 1 - 4.80e9T + 1.55e19T^{2} \)
89 \( 1 + 8.47e9T + 3.11e19T^{2} \)
97 \( 1 - 4.64e9T + 7.37e19T^{2} \)
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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

−15.75643571516863025618335615962, −15.38036351399585365241457219829, −14.13525991761727283463816635028, −12.47823042989562198888270279484, −10.96749378802414293340326698724, −9.507144493462758490186130568525, −7.77551000392208237775697055375, −6.34635516149990237053827832832, −5.37146247916115346546107474157, −2.83757041789778197688592175687, 0.35978871445558782216830727242, 1.28978813394271911508373477305, 4.07052610279364363480277498685, 5.07110591073221669671885843186, 7.67857173265901222085642748453, 9.324383245727129177802504767854, 10.54331992840190541915017332613, 11.80175293159072253886217396901, 13.09849995660811446697168663825, 13.71913418907754183047255595171

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