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Properties

Label 1-3024-3024.1685-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.891 + 0.452i$
Analytic cond. $14.0433$
Root an. cond. $14.0433$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.766 + 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (−0.173 + 0.984i)31-s + i·37-s + (−0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (0.173 + 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)5-s + (0.984 − 0.173i)11-s + (0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.766 + 0.642i)23-s + (0.939 + 0.342i)25-s + (−0.984 + 0.173i)29-s + (−0.173 + 0.984i)31-s + i·37-s + (−0.173 + 0.984i)41-s + (0.642 + 0.766i)43-s + (0.173 + 0.984i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.891 + 0.452i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.891 + 0.452i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.287152381 + 0.5470791094i\)
\(L(\frac12)\) \(\approx\) \(2.287152381 + 0.5470791094i\)
\(L(1)\) \(\approx\) \(1.396681530 + 0.1078954584i\)
\(L(1)\) \(\approx\) \(1.396681530 + 0.1078954584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (0.984 - 0.173i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.984 + 0.173i)T \)
31 \( 1 + (-0.173 + 0.984i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.642 - 0.766i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (0.984 - 0.173i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.93954885102084379832207231706, −18.27272224121282922443746906562, −17.52495817099018648072887515374, −16.86238684774544697543809043401, −16.55069225413032000503631287258, −15.27732951592373353092303464172, −14.87494509959233443826488626840, −14.02286865124474484111806581940, −13.35879863318797388206246841423, −12.76177787886627923481780753104, −12.07357804072487385799419581592, −10.908656619059439650677653067599, −10.65702240016272427715637742855, −9.64984360839381615695218629699, −8.8921732805488475200878993430, −8.56049876703814539091427649081, −7.36642651951080990690755103159, −6.47858318275967653231648544693, −6.01776718445007829472669531266, −5.27940868575455216399657947978, −4.10096008435047745108368984259, −3.71929111835075569861044266501, −2.29603163560065537976554313653, −1.829230351630851318255056345399, −0.7999588994807226598001457573, 1.075912314543581209271720020942, 1.71648667277341253091244565915, 2.7509446856239841781625807190, 3.49178245959835969004193348122, 4.51356802673953276924882085605, 5.24027054080292089687720684997, 6.28707667860934957516531897725, 6.53292973530403468212641862306, 7.45209834477226382250488569759, 8.60421540841056161504690845799, 9.18460516631528194031924328645, 9.608933061931681779245147053161, 10.8417658753101877093205879118, 11.06738215878790050850355154006, 11.99973734207326226143289062667, 13.02062581882463238157777469720, 13.45270078338185863463340537440, 14.14259896828202598358540438692, 14.81076880859707257154698188221, 15.59096340092373082725380749608, 16.45217856940925384593798882094, 17.06976153308809477980788886765, 17.676078039873588961758358251149, 18.37819630031475892276925140803, 18.98836559955637723594881470602

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