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Properties

Label 2-24-8.5-c7-0-1
Degree $2$
Conductor $24$
Sign $-0.623 + 0.781i$
Analytic cond. $7.49724$
Root an. cond. $2.73810$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.69 + 8.29i)2-s + 27i·3-s + (−9.58 − 127. i)4-s + 455. i·5-s + (−223. − 207. i)6-s − 743.·7-s + (1.13e3 + 902. i)8-s − 729·9-s + (−3.77e3 − 3.50e3i)10-s − 5.47e3i·11-s + (3.44e3 − 258. i)12-s − 6.21e3i·13-s + (5.72e3 − 6.16e3i)14-s − 1.22e4·15-s + (−1.62e4 + 2.44e3i)16-s − 2.63e4·17-s + ⋯
L(s)  = 1  + (−0.680 + 0.733i)2-s + 0.577i·3-s + (−0.0748 − 0.997i)4-s + 1.62i·5-s + (−0.423 − 0.392i)6-s − 0.819·7-s + (0.781 + 0.623i)8-s − 0.333·9-s + (−1.19 − 1.10i)10-s − 1.24i·11-s + (0.575 − 0.0432i)12-s − 0.785i·13-s + (0.557 − 0.600i)14-s − 0.940·15-s + (−0.988 + 0.149i)16-s − 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.623 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.623 + 0.781i$
Analytic conductor: \(7.49724\)
Root analytic conductor: \(2.73810\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :7/2),\ -0.623 + 0.781i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.170018 - 0.352948i\)
\(L(\frac12)\) \(\approx\) \(0.170018 - 0.352948i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.69 - 8.29i)T \)
3 \( 1 - 27iT \)
good5 \( 1 - 455. iT - 7.81e4T^{2} \)
7 \( 1 + 743.T + 8.23e5T^{2} \)
11 \( 1 + 5.47e3iT - 1.94e7T^{2} \)
13 \( 1 + 6.21e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.63e4T + 4.10e8T^{2} \)
19 \( 1 - 2.33e4iT - 8.93e8T^{2} \)
23 \( 1 - 5.09e4T + 3.40e9T^{2} \)
29 \( 1 - 1.85e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.84e5T + 2.75e10T^{2} \)
37 \( 1 - 2.35e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.39e5T + 1.94e11T^{2} \)
43 \( 1 + 3.04e5iT - 2.71e11T^{2} \)
47 \( 1 - 9.23e5T + 5.06e11T^{2} \)
53 \( 1 - 1.21e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.07e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.76e5iT - 3.14e12T^{2} \)
67 \( 1 + 2.56e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.50e6T + 9.09e12T^{2} \)
73 \( 1 + 3.68e6T + 1.10e13T^{2} \)
79 \( 1 + 3.93e5T + 1.92e13T^{2} \)
83 \( 1 - 2.77e6iT - 2.71e13T^{2} \)
89 \( 1 + 9.69e6T + 4.42e13T^{2} \)
97 \( 1 - 9.83e6T + 8.07e13T^{2} \)
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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

−16.79214780478813965173637690390, −15.65711153118877096346935149480, −14.81391273623779667686740136927, −13.62945898662348007252230153265, −10.98932242238664304281881788078, −10.36396977605274413222978430503, −8.836845551535806474317344333924, −7.05584744742849632703493617629, −5.88789838715415080441995117582, −3.16466104258851714977611655734, 0.24793895676006740432313063532, 1.94126982389817703782124886133, 4.48033044127846278033921190528, 7.05656551291368284395415863526, 8.762670746597119490024378822388, 9.559502640399605867745831240135, 11.56938379910158165632201501826, 12.75842207663140142061576044473, 13.23002316268990569867812349737, 15.77302284381033518333459082332

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