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Properties

Label 2-24-8.5-c7-0-4
Degree $2$
Conductor $24$
Sign $0.136 - 0.990i$
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  + (−6.09 + 9.53i)2-s − 27i·3-s + (−53.6 − 116. i)4-s + 137. i·5-s + (257. + 164. i)6-s + 808.·7-s + (1.43e3 + 196. i)8-s − 729·9-s + (−1.30e3 − 836. i)10-s + 8.24e3i·11-s + (−3.13e3 + 1.44e3i)12-s + 3.40e3i·13-s + (−4.92e3 + 7.70e3i)14-s + 3.70e3·15-s + (−1.06e4 + 1.24e4i)16-s + 2.07e4·17-s + ⋯
L(s)  = 1  + (−0.538 + 0.842i)2-s − 0.577i·3-s + (−0.419 − 0.907i)4-s + 0.490i·5-s + (0.486 + 0.311i)6-s + 0.890·7-s + (0.990 + 0.136i)8-s − 0.333·9-s + (−0.413 − 0.264i)10-s + 1.86i·11-s + (−0.524 + 0.242i)12-s + 0.430i·13-s + (−0.479 + 0.750i)14-s + 0.283·15-s + (−0.648 + 0.761i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.136 - 0.990i$
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: \((2,\ 24,\ (\ :7/2),\ 0.136 - 0.990i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.938252 + 0.818251i\)
\(L(\frac12)\) \(\approx\) \(0.938252 + 0.818251i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.09 - 9.53i)T \)
3 \( 1 + 27iT \)
good5 \( 1 - 137. iT - 7.81e4T^{2} \)
7 \( 1 - 808.T + 8.23e5T^{2} \)
11 \( 1 - 8.24e3iT - 1.94e7T^{2} \)
13 \( 1 - 3.40e3iT - 6.27e7T^{2} \)
17 \( 1 - 2.07e4T + 4.10e8T^{2} \)
19 \( 1 - 6.30e3iT - 8.93e8T^{2} \)
23 \( 1 - 6.88e4T + 3.40e9T^{2} \)
29 \( 1 - 4.20e4iT - 1.72e10T^{2} \)
31 \( 1 + 2.14e5T + 2.75e10T^{2} \)
37 \( 1 + 1.82e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.95e5T + 1.94e11T^{2} \)
43 \( 1 - 4.33e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.32e6T + 5.06e11T^{2} \)
53 \( 1 + 2.25e5iT - 1.17e12T^{2} \)
59 \( 1 - 2.06e6iT - 2.48e12T^{2} \)
61 \( 1 + 3.11e6iT - 3.14e12T^{2} \)
67 \( 1 + 1.22e6iT - 6.06e12T^{2} \)
71 \( 1 + 1.63e6T + 9.09e12T^{2} \)
73 \( 1 - 3.43e6T + 1.10e13T^{2} \)
79 \( 1 - 6.60e6T + 1.92e13T^{2} \)
83 \( 1 - 3.38e6iT - 2.71e13T^{2} \)
89 \( 1 + 6.88e5T + 4.42e13T^{2} \)
97 \( 1 + 3.48e6T + 8.07e13T^{2} \)
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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

−16.63146004086350012453555241350, −14.87002131280323795100073786135, −14.46622635828389131363138659086, −12.67410746293441246735013822601, −10.97464268427021941874282269786, −9.470467509884125873483580643050, −7.79063886224520239423289497335, −6.87525196985877590330456704045, −4.98155596542836114764692033432, −1.62880546423282170437994149029, 0.899366050337093792132738692411, 3.30199613959554828324084689747, 5.15815607958807458889244412642, 8.079163888280267061853217900299, 9.048752865130291214321441151379, 10.68078920785777964144154300765, 11.52499123842810666644880028090, 13.08762605071516290460794940395, 14.45417748858904629306889295967, 16.28654930860203236917006648694

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