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Properties

Label 1-3024-3024.1843-r1-0-0
Degree $1$
Conductor $3024$
Sign $-0.387 + 0.921i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.387 + 0.921i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ -0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6310391799 + 0.9497507286i\)
\(L(\frac12)\) \(\approx\) \(0.6310391799 + 0.9497507286i\)
\(L(1)\) \(\approx\) \(1.077737250 + 0.01141307488i\)
\(L(1)\) \(\approx\) \(1.077737250 + 0.01141307488i\)

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

−18.5278009503052531745230659399, −17.91560830337995828114301919211, −17.41594576155712841091607168765, −16.63960185500128091324494023635, −15.857438201020807878373821465469, −14.95659468299553805623076970612, −14.55992088782544267239890018812, −13.673027088040189145720028296582, −13.26054100588872139899341531299, −12.24417716960583053780867014633, −11.511187695208220957083279018229, −10.82266556054122898961332857035, −10.05058223756801649491499975370, −9.58451631194567687368224826940, −8.53971022288083320094441460883, −7.936701674806736362352538177841, −6.88198189727575245037242125191, −6.3811872216029473815511884271, −5.631226659505838116544263193139, −4.82534970449068145038810900329, −3.70198644751751924066629189672, −3.057106536079412659990721694088, −2.228986560129190364007998896917, −1.30748648070075627695781264476, −0.17999982254251482833825302529, 0.98482859903291232028866242590, 1.86129676184931260532250309583, 2.45037826243324535772912187650, 3.78995466264199775358880329320, 4.44406929249846451320443922847, 5.2485819393122874167920713635, 5.87255013602368392603596917972, 6.97311853133829767466257975447, 7.35631645212703141789691498567, 8.54401520961196433973461164675, 9.18935895914472915722913319131, 9.719158879529976114463105501340, 10.34090003021880188176419474159, 11.62454856319130326788889059254, 11.97000573396586271285791187303, 12.71341239610329669443789592067, 13.53627656713030948435423549084, 14.204753864476114407560894447200, 14.6673581402104583743449005914, 15.88785595969048853153855533620, 16.246354006196722036903437888782, 17.051780638816371908096630676723, 17.701271544352429481771933237189, 18.188202086805046462831058042572, 19.157625741683285675428605932892

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