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Properties

Label 1-21-21.5-r0-0-0
Degree $1$
Conductor $21$
Sign $0.605 + 0.795i$
Analytic cond. $0.0975235$
Root an. cond. $0.0975235$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.0975235\)
Root analytic conductor: \(0.0975235\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 21,\ (0:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6694938424 + 0.3318863448i\)
\(L(\frac12)\) \(\approx\) \(0.6694938424 + 0.3318863448i\)
\(L(1)\) \(\approx\) \(0.9424309741 + 0.3602888690i\)
\(L(1)\) \(\approx\) \(0.9424309741 + 0.3602888690i\)

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

−39.30510211146014997278007439712, −38.48269784891810374623718432223, −37.35275904908157588671755386615, −35.89853130034376326083308420766, −34.21435508975330046908853886670, −32.833383076707142705541313466708, −31.32619916618084365232408180972, −30.44017502751045189140085211736, −29.21052816771762619536447304847, −27.72055470111870522019971924738, −26.538738063046226704313142259794, −24.46857275710259925298939854837, −22.86190794944307696502341998623, −22.1242055806246711510240906359, −20.33203324297432567727436708573, −19.21531180032017935354707530881, −17.8099935932126404559635957513, −15.31430467617490589662058340581, −14.16776444266309741666574815350, −12.340493546549576107432982097443, −11.07663219778130032536446790237, −9.535060568676000726486421312049, −6.963051664297083192261294811440, −4.61676697612220809176794508841, −2.713661054530821908489531505200, 3.93679131959732380659178331816, 5.63227241518595499288917969016, 7.63347520175472358408746225190, 9.05833888683324419147075977365, 11.8376479720516422789104848514, 13.19241251818307983248006908592, 14.8119131191553031284187735102, 16.25833267856615512172453481886, 17.2924857541050276153024313542, 19.378901146721629871300690671900, 21.13548794719424623266817352206, 22.53845428531322685194775289527, 24.05665419571425407988859176941, 24.76209333329044632231109900533, 26.5239049066800223364054045728, 27.66619023623240865386270687932, 29.54949492304203645567746298403, 31.23783466594073402673008418062, 32.093169105228246916449007509745, 33.28830860361225655819214834053, 34.83684405639583132685959043978, 35.64257411497973349963621131466, 37.12049190455360556345390151354, 39.245778385534817742539824247568, 39.9757841221087542132090392787

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