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Properties

Label 1-21-21.2-r1-0-0
Degree $1$
Conductor $21$
Sign $-0.0633 + 0.997i$
Analytic cond. $2.25676$
Root an. cond. $2.25676$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s − 20-s + 22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $-0.0633 + 0.997i$
Analytic conductor: \(2.25676\)
Root analytic conductor: \(2.25676\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 21,\ (1:\ ),\ -0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119373288 + 1.192660627i\)
\(L(\frac12)\) \(\approx\) \(1.119373288 + 1.192660627i\)
\(L(1)\) \(\approx\) \(1.141608742 + 0.7593774607i\)
\(L(1)\) \(\approx\) \(1.141608742 + 0.7593774607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + T \)
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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

−39.189747011792309352002046788789, −37.93560052427407836388259041719, −36.697864474498195969919747577484, −35.613497094931050093311998530581, −33.32383167552013224190719473763, −32.484204295731009453098836670545, −31.10642269186750944554083916492, −29.863580418798960306799522844544, −28.5029282750993406787087103555, −27.69053131480802296539058343815, −25.5147724933501927559619647108, −24.00718281745723516950085479751, −22.666672209770741748167873251, −21.12402284421267552356689866092, −20.26899064118222038336848795427, −18.6285197154408709494386107626, −16.95697615757441970894706927005, −14.87263645319252306374596971776, −13.28402633710605058592580067155, −12.17774973539419563968302301124, −10.329513358619587930526372948162, −8.84449363843428597306641234141, −5.8751261712862967192297656850, −4.13202204632408647064707313865, −1.61201939105444709858788997003, 3.38123798130451827202973386396, 5.70086521677533954748123778311, 7.08050476257602177700061314530, 9.02729596205879432665264889066, 11.25009364810249046204992106384, 13.35979207058571223825215003508, 14.41833423973691722484914262443, 15.92053869855272052388183740450, 17.45313784141581656464295650630, 18.78076149692662465730077844967, 21.17602198380544411305641449277, 22.29173374741619256524879198176, 23.54291842636638356773863602664, 25.09182551808548267340125569260, 26.09555313814283836834102187054, 27.40042617941045636858938267248, 29.59888903169292550313337820430, 30.6363221120543385600721457223, 32.140109237214601411081419729094, 33.31808790813385758521621654512, 34.36460321311181453481080963285, 35.5280025520395201455620714587, 37.14319148227260714163461378756, 38.567990239189238876830991233428, 40.23685401838672866861432820238

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