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Properties

Label 1-3024-3024.1789-r0-0-0
Degree $1$
Conductor $3024$
Sign $-0.713 + 0.700i$
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 + (−0.642 + 0.766i)13-s + 17-s + i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.642 + 0.766i)29-s + (−0.939 + 0.342i)31-s + (−0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.939 − 0.342i)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 + (−0.642 + 0.766i)13-s + 17-s + i·19-s + (−0.766 − 0.642i)23-s + (0.939 + 0.342i)25-s + (0.642 + 0.766i)29-s + (−0.939 + 0.342i)31-s + (−0.866 + 0.5i)37-s + (−0.766 − 0.642i)41-s + (−0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.866 + 0.5i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.713 + 0.700i$
Analytic conductor: \(14.0433\)
Root analytic conductor: \(14.0433\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1789, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (0:\ ),\ -0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2251200134 + 0.5507538858i\)
\(L(\frac12)\) \(\approx\) \(0.2251200134 + 0.5507538858i\)
\(L(1)\) \(\approx\) \(0.8039721423 + 0.09104548181i\)
\(L(1)\) \(\approx\) \(0.8039721423 + 0.09104548181i\)

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 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + T \)
19 \( 1 + iT \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (0.642 - 0.766i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (0.984 + 0.173i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.642 + 0.766i)T \)
89 \( 1 - T \)
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

−18.99956478587177070747080677992, −18.02441752405462088677876592055, −17.41469179193330137756427806512, −16.691569481435548396580483015336, −15.92228930859066986984160891919, −15.26462021952762408134024884231, −14.68400058617411914230188057899, −14.02877653639399137149627573585, −13.05631685973235079462492499937, −12.27582914683238370834712159890, −11.76470464866158104198035729550, −11.15862741465004195228416476432, −10.16958508563948624087973939027, −9.613700374070372382735465816021, −8.636139709652171327679827038157, −7.957039151619145572652733742704, −7.260830499186184477715541873072, −6.65043220396835401025334080767, −5.58652538061497600133565522098, −4.84826891785507116491817690323, −3.911590697799068275326887213114, −3.377450708748086612246091979339, −2.43336433157294527030293611102, −1.28439713705441627338906628163, −0.20138339370584897889531168188, 1.16831043413441951195508178881, 1.96001700750214849101256060535, 3.345872341481936518539438265263, 3.700625217956839761444628101646, 4.64199830057428379350346313438, 5.32667231677150066995065118827, 6.50248244344646645066323749688, 6.94820623638457355238695349259, 7.97458828443078258957093162291, 8.39776226183721115019128397225, 9.33836511423793551557755887347, 10.00512799847864718256511388150, 10.91145838110380162150229627112, 11.69092454810674186778233559448, 12.27637121428158980366541944526, 12.60286599834101704365207625798, 14.095914520427944161051094540870, 14.28409727206997487921471951884, 15.03670434991494941292118380108, 15.970827637823295226111116189653, 16.62236126019262729735317189365, 16.87822049408730587795549643793, 18.03523562037922879242577511148, 18.78502720169259453153472803792, 19.2917722178676175143781819043

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