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Properties

Label 2-24-24.11-c7-0-15
Degree $2$
Conductor $24$
Sign $0.822 - 0.568i$
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  + (−3.79 + 10.6i)2-s + (46.6 − 2.79i)3-s + (−99.2 − 80.8i)4-s + 395.·5-s + (−147. + 508. i)6-s − 1.39e3i·7-s + (1.23e3 − 750. i)8-s + (2.17e3 − 261. i)9-s + (−1.49e3 + 4.21e3i)10-s + 5.39e3i·11-s + (−4.85e3 − 3.49e3i)12-s + 3.92e3i·13-s + (1.48e4 + 5.29e3i)14-s + (1.84e4 − 1.10e3i)15-s + (3.30e3 + 1.60e4i)16-s − 5.90e3i·17-s + ⋯
L(s)  = 1  + (−0.335 + 0.942i)2-s + (0.998 − 0.0598i)3-s + (−0.775 − 0.631i)4-s + 1.41·5-s + (−0.278 + 0.960i)6-s − 1.53i·7-s + (0.855 − 0.518i)8-s + (0.992 − 0.119i)9-s + (−0.474 + 1.33i)10-s + 1.22i·11-s + (−0.811 − 0.584i)12-s + 0.495i·13-s + (1.44 + 0.515i)14-s + (1.41 − 0.0845i)15-s + (0.201 + 0.979i)16-s − 0.291i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(7.49724\)
Root analytic conductor: \(2.73810\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :7/2),\ 0.822 - 0.568i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.10192 + 0.655864i\)
\(L(\frac12)\) \(\approx\) \(2.10192 + 0.655864i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.79 - 10.6i)T \)
3 \( 1 + (-46.6 + 2.79i)T \)
good5 \( 1 - 395.T + 7.81e4T^{2} \)
7 \( 1 + 1.39e3iT - 8.23e5T^{2} \)
11 \( 1 - 5.39e3iT - 1.94e7T^{2} \)
13 \( 1 - 3.92e3iT - 6.27e7T^{2} \)
17 \( 1 + 5.90e3iT - 4.10e8T^{2} \)
19 \( 1 - 1.48e4T + 8.93e8T^{2} \)
23 \( 1 + 7.36e4T + 3.40e9T^{2} \)
29 \( 1 - 2.65e4T + 1.72e10T^{2} \)
31 \( 1 + 2.05e5iT - 2.75e10T^{2} \)
37 \( 1 - 1.72e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.41e5iT - 1.94e11T^{2} \)
43 \( 1 + 2.79e5T + 2.71e11T^{2} \)
47 \( 1 + 4.18e5T + 5.06e11T^{2} \)
53 \( 1 + 4.27e5T + 1.17e12T^{2} \)
59 \( 1 + 1.65e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.02e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.38e6T + 6.06e12T^{2} \)
71 \( 1 + 8.22e5T + 9.09e12T^{2} \)
73 \( 1 + 1.84e6T + 1.10e13T^{2} \)
79 \( 1 - 8.46e5iT - 1.92e13T^{2} \)
83 \( 1 - 1.30e5iT - 2.71e13T^{2} \)
89 \( 1 - 4.39e6iT - 4.42e13T^{2} \)
97 \( 1 - 3.64e6T + 8.07e13T^{2} \)
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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

−16.36279327129255903611906136200, −14.76300633393141495855597428950, −13.86327594501313088509673165613, −13.24437067928604067924233483241, −10.02752427958678968086997136348, −9.634185964113816498096163075896, −7.76874548913224676205964098367, −6.62109589966609971988276351161, −4.44639522456366638797063604494, −1.58797100913741317151310122301, 1.84934758122825797645731350204, 3.03653188696796342666807100949, 5.61968667799847054526342926118, 8.413012333120714245376378230907, 9.227066792063078797608178501447, 10.37635115598158118295728638030, 12.23538806669800814138511606651, 13.44070969901681874828943236224, 14.31260287175361100122396902904, 16.01850625775876428426586352408

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