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Properties

Label 1-3024-3024.1733-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  + (0.984 − 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

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

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: $\chi_{3024} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.952169688 + 1.297072507i\)
\(L(\frac12)\) \(\approx\) \(1.952169688 + 1.297072507i\)
\(L(1)\) \(\approx\) \(1.245518269 + 0.05332344035i\)
\(L(1)\) \(\approx\) \(1.245518269 + 0.05332344035i\)

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

−18.835737831419139857299926546102, −17.9241795388626907163671377322, −17.45163763295594286897546646099, −16.78668452291842626433524482395, −15.90269072851215762046930606925, −15.23858455468840864854703828174, −14.563315484343796452554302800662, −13.648359517113147393025720765551, −13.21815580885608539856994857968, −12.6279201733431105914375257618, −11.59826319859812772338206203148, −10.58617322464835127949526833621, −10.49271151111619490188663508105, −9.451281583692461487673524026264, −8.75670689262829473471202635245, −8.03933295899620243475861229104, −7.09394839546321732232854298419, −6.309161528404593152513335740081, −5.68933998320921325340203360618, −4.97508921934821830504318580459, −3.97333884836314682300187318721, −3.06844182991732432518302024762, −2.20995321195127190403941599594, −1.5156335443792940254613016082, −0.38030456713562795893610897696, 0.93224114197888245361850711708, 1.582977604031975015738580934538, 2.81947681633793086893727249334, 3.110635721143853265455111070403, 4.66181240843242735184335850105, 4.97969386000937793779430970804, 6.14976466395608718194431510748, 6.35231960762235853659357385204, 7.55437570116104630274622452148, 8.35431317787746848640251431817, 8.9316903418873835739688302238, 9.83207217924564729172279281114, 10.50202404262555012363020695698, 10.96507276148149214455037077017, 12.08407947984734127794330631968, 12.799253796783179388560770067298, 13.39458950611483628099373070406, 13.98658654571846108722136369516, 14.72439354111224003111977043835, 15.591938460745308526087015423665, 16.34180317377736251931156898739, 16.800449690114885396590093863084, 17.77841713828374534862585129164, 18.275192951497451209012238611898, 18.77153123080000793792524013139

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