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Properties

Label 1-21e2-441.299-r0-0-0
Degree $1$
Conductor $441$
Sign $0.713 + 0.700i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.623 − 0.781i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.222 − 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (0.623 − 0.781i)5-s + (−0.623 + 0.781i)8-s + (0.988 − 0.149i)10-s + (0.222 − 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (0.0747 − 0.997i)17-s + (0.5 + 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.713 + 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.098906565 + 0.8584671705i\)
\(L(\frac12)\) \(\approx\) \(2.098906565 + 0.8584671705i\)
\(L(1)\) \(\approx\) \(1.647875887 + 0.5210228064i\)
\(L(1)\) \(\approx\) \(1.647875887 + 0.5210228064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.733 + 0.680i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.733 + 0.680i)T \)
17 \( 1 + (0.0747 - 0.997i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.826 - 0.563i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 + 0.563i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (-0.0747 + 0.997i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.955 - 0.294i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.733 + 0.680i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−23.73869825555758270955732193263, −22.70666743637821427631051346125, −22.43773931113139118747861297304, −21.37799173516225526960313315487, −20.69007409863953600771610508685, −19.80534356263897985450972123343, −18.890176865447111322550331732831, −18.04060965537821810904464353586, −17.30288153159072454262759411690, −15.77191891871264424868480655340, −14.91397146386256818616962021685, −14.40075441034633604075735069898, −13.15079414617212383615076328645, −12.82588464542391821217954699752, −11.416062858328597586577549871698, −10.784887493305895575518467250416, −9.921674014429144321480234652243, −9.074662935846192486257740443722, −7.43856306534422249178336404870, −6.42095304714696145289399665149, −5.6378687126689125889983168919, −4.487376294494888683024339089089, −3.36259168856014694586704141041, −2.44067538379052828196146988773, −1.34429523494125158468344035469, 1.33025003600244416836167909669, 2.8776521004645650898563882531, 3.95860359607414551649237777261, 5.05946904990776306219869357657, 5.82757348860501440540911230871, 6.68489536721407058040644997683, 7.92915790093331295915279021959, 8.805961425093436271123291968252, 9.59163339376762216416217595781, 11.21144771972224826418506892911, 11.93576321350875262786101326626, 13.05046987387350303975866967251, 13.719582834710261400344387294563, 14.297278974130816742150002848568, 15.59528929792792018642568653570, 16.44097750293449763540927209800, 16.82294502702812457720870077542, 17.94919753963339447181923883737, 18.84500046787095709177795758656, 20.244438999170411033665056629478, 21.03227813680714723171915987531, 21.52309009744165311851695216461, 22.55180531564428641782432779942, 23.40687438851935688634767867183, 24.233609204265216341023342680754

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