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Properties

Label 2-24-8.5-c9-0-4
Degree $2$
Conductor $24$
Sign $-0.746 + 0.665i$
Analytic cond. $12.3608$
Root an. cond. $3.51580$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.44 + 21.9i)2-s + 81i·3-s + (−452. + 239. i)4-s + 1.82e3i·5-s + (−1.77e3 + 440. i)6-s + 2.18e3·7-s + (−7.71e3 − 8.64e3i)8-s − 6.56e3·9-s + (−4.00e4 + 9.92e3i)10-s + 4.68e4i·11-s + (−1.93e4 − 3.66e4i)12-s − 1.16e5i·13-s + (1.18e4 + 4.79e4i)14-s − 1.47e5·15-s + (1.47e5 − 2.16e5i)16-s − 8.06e4·17-s + ⋯
L(s)  = 1  + (0.240 + 0.970i)2-s + 0.577i·3-s + (−0.884 + 0.466i)4-s + 1.30i·5-s + (−0.560 + 0.138i)6-s + 0.343·7-s + (−0.665 − 0.746i)8-s − 0.333·9-s + (−1.26 + 0.313i)10-s + 0.964i·11-s + (−0.269 − 0.510i)12-s − 1.12i·13-s + (0.0826 + 0.333i)14-s − 0.753·15-s + (0.564 − 0.825i)16-s − 0.234·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(12.3608\)
Root analytic conductor: \(3.51580\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :9/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.470475 - 1.23380i\)
\(L(\frac12)\) \(\approx\) \(0.470475 - 1.23380i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.44 - 21.9i)T \)
3 \( 1 - 81iT \)
good5 \( 1 - 1.82e3iT - 1.95e6T^{2} \)
7 \( 1 - 2.18e3T + 4.03e7T^{2} \)
11 \( 1 - 4.68e4iT - 2.35e9T^{2} \)
13 \( 1 + 1.16e5iT - 1.06e10T^{2} \)
17 \( 1 + 8.06e4T + 1.18e11T^{2} \)
19 \( 1 + 3.07e5iT - 3.22e11T^{2} \)
23 \( 1 + 5.23e5T + 1.80e12T^{2} \)
29 \( 1 - 6.80e6iT - 1.45e13T^{2} \)
31 \( 1 + 3.21e6T + 2.64e13T^{2} \)
37 \( 1 - 1.31e7iT - 1.29e14T^{2} \)
41 \( 1 - 3.40e7T + 3.27e14T^{2} \)
43 \( 1 - 4.20e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.78e7T + 1.11e15T^{2} \)
53 \( 1 + 2.42e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.94e7iT - 8.66e15T^{2} \)
61 \( 1 - 6.35e7iT - 1.16e16T^{2} \)
67 \( 1 + 2.05e8iT - 2.72e16T^{2} \)
71 \( 1 - 1.01e8T + 4.58e16T^{2} \)
73 \( 1 - 3.63e8T + 5.88e16T^{2} \)
79 \( 1 + 4.14e8T + 1.19e17T^{2} \)
83 \( 1 - 1.94e8iT - 1.86e17T^{2} \)
89 \( 1 - 8.66e8T + 3.50e17T^{2} \)
97 \( 1 + 2.67e8T + 7.60e17T^{2} \)
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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

−16.07919908797741436795091125599, −14.95748598788541505512512008464, −14.48271855912014752611104969491, −12.83322599963755234140029434358, −10.96791028318659073858890452179, −9.663439604449783074296668346315, −7.88277813077143449543876636665, −6.58788851087752632741708115847, −4.91356854859587209605635768216, −3.16366189651288382243067385638, 0.53834962095257919669315738789, 1.90020866654344701748720782043, 4.17495207408165485213881485856, 5.73291949586180039477224645852, 8.274374214046128398390762359167, 9.327546990864956617995822284376, 11.24007724736114095311822343319, 12.22868672093230548514101848125, 13.31529545932519507091095099637, 14.27428184340948624235182365553

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