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Properties

Label 2-24-8.3-c6-0-7
Degree $2$
Conductor $24$
Sign $0.970 - 0.241i$
Analytic cond. $5.52129$
Root an. cond. $2.34974$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.97 − 0.648i)2-s + 15.5·3-s + (63.1 − 10.3i)4-s + 232. i·5-s + (124. − 10.1i)6-s − 483. i·7-s + (496. − 123. i)8-s + 243·9-s + (150. + 1.85e3i)10-s − 538.·11-s + (984. − 161. i)12-s − 764. i·13-s + (−313. − 3.85e3i)14-s + 3.62e3i·15-s + (3.88e3 − 1.30e3i)16-s − 4.27e3·17-s + ⋯
L(s)  = 1  + (0.996 − 0.0811i)2-s + 0.577·3-s + (0.986 − 0.161i)4-s + 1.85i·5-s + (0.575 − 0.0468i)6-s − 1.41i·7-s + (0.970 − 0.241i)8-s + 0.333·9-s + (0.150 + 1.85i)10-s − 0.404·11-s + (0.569 − 0.0933i)12-s − 0.347i·13-s + (−0.114 − 1.40i)14-s + 1.07i·15-s + (0.947 − 0.319i)16-s − 0.869·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(24\)    =    \(2^{3} \cdot 3\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(5.52129\)
Root analytic conductor: \(2.34974\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{24} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 24,\ (\ :3),\ 0.970 - 0.241i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.13677 + 0.383982i\)
\(L(\frac12)\) \(\approx\) \(3.13677 + 0.383982i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-7.97 + 0.648i)T \)
3 \( 1 - 15.5T \)
good5 \( 1 - 232. iT - 1.56e4T^{2} \)
7 \( 1 + 483. iT - 1.17e5T^{2} \)
11 \( 1 + 538.T + 1.77e6T^{2} \)
13 \( 1 + 764. iT - 4.82e6T^{2} \)
17 \( 1 + 4.27e3T + 2.41e7T^{2} \)
19 \( 1 + 5.00e3T + 4.70e7T^{2} \)
23 \( 1 + 1.26e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.61e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.63e4iT - 8.87e8T^{2} \)
37 \( 1 + 4.60e4iT - 2.56e9T^{2} \)
41 \( 1 - 5.31e4T + 4.75e9T^{2} \)
43 \( 1 - 8.43e4T + 6.32e9T^{2} \)
47 \( 1 - 1.15e5iT - 1.07e10T^{2} \)
53 \( 1 + 1.58e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.20e3T + 4.21e10T^{2} \)
61 \( 1 - 3.04e5iT - 5.15e10T^{2} \)
67 \( 1 - 1.44e5T + 9.04e10T^{2} \)
71 \( 1 + 1.59e4iT - 1.28e11T^{2} \)
73 \( 1 + 2.61e5T + 1.51e11T^{2} \)
79 \( 1 - 3.37e5iT - 2.43e11T^{2} \)
83 \( 1 + 4.61e5T + 3.26e11T^{2} \)
89 \( 1 - 1.14e6T + 4.96e11T^{2} \)
97 \( 1 + 2.88e5T + 8.32e11T^{2} \)
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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

−15.97990271433783550626063402687, −14.70630476994027749553686384696, −14.14532602399746766583606032785, −12.99337160143517613610506285202, −10.87865519256639698454603653107, −10.46989140461554878158969581956, −7.49388811473345455406512620371, −6.59518291676428518227722601495, −4.00279449224619027091359390939, −2.62738961531979000715048795330, 2.09546711852869332708797141474, 4.45543989801610603635149381114, 5.75488884286979522556023573770, 8.149066993404068169138898313625, 9.256395342538905309066154755379, 11.72090216884428394885535639744, 12.71060368244155660034765790193, 13.52591362060427854181401346147, 15.27338255108040390917644971196, 15.88362683574882369642774811272

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