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Properties

Label 1-3024-3024.1381-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: $\chi_{3024} (1381, \cdot )$
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

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

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