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Properties

Label 1-3024-3024.2707-r0-0-0
Degree $1$
Conductor $3024$
Sign $0.310 + 0.950i$
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  + (−0.642 + 0.766i)5-s + (0.642 + 0.766i)11-s + (0.642 − 0.766i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.642 + 0.766i)29-s + (0.766 + 0.642i)31-s − i·37-s + (0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)5-s + (0.642 + 0.766i)11-s + (0.642 − 0.766i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (−0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (0.642 + 0.766i)29-s + (0.766 + 0.642i)31-s − i·37-s + (0.766 + 0.642i)41-s + (0.342 − 0.939i)43-s + (0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.310 + 0.950i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (2707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.201505466 + 0.8714143076i\)
\(L(\frac12)\) \(\approx\) \(1.201505466 + 0.8714143076i\)
\(L(1)\) \(\approx\) \(0.9980608594 + 0.2326747333i\)
\(L(1)\) \(\approx\) \(0.9980608594 + 0.2326747333i\)

Euler product

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

−18.9774288154522734060645911537, −18.4073305275905311890264587830, −17.32163949380175791930675627591, −16.69956428621615729596255471031, −16.20081929786213328386551141561, −15.6003135590174000753585828438, −14.687607244454971104288793287935, −13.87155127217500841105189554002, −13.45294961018558628601384318275, −12.32492556170869254492682953491, −11.916870714170834691813970272230, −11.30075246932628424429946335045, −10.4256513078765106506535434203, −9.44839271376563838115352541075, −8.87972462520687715940554119696, −8.1452792467136823171864706981, −7.59525624498149762815722768527, −6.33995120665934425262333000693, −6.04477239628139952087500288444, −4.81126441372499548385086232307, −4.2011649071346024327924870709, −3.57470084985236443212781219660, −2.48888231791822116400661003357, −1.38580187386974420417266179046, −0.586355232037839805611581456629, 0.92620975274241028200618880847, 2.01692019495758505886663474654, 2.90002301795756582584329293049, 3.829547992300240195139808167093, 4.22969982354711633769703837667, 5.41135316879653144505553663496, 6.284963224299671116427382125765, 6.856618023429667818498800244158, 7.70975861641297714379963945626, 8.327101102063556879262111986848, 9.126386836797033704648031046896, 10.21596686683641968024857333018, 10.58526547811990400607020043957, 11.33719632420492169302136039664, 12.3308694512931763389163117246, 12.53000223581640030519795535550, 13.75001734687605830451614110951, 14.32844437618056665830011750952, 15.144325173193361214364811554516, 15.48425005326976417335101484872, 16.31009393891902336183463083355, 17.25577955349730226960681965734, 17.8254693048682949549468926453, 18.430935631738903958398984693800, 19.36497916092304599931415162808

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