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Properties

Label 1-21e2-441.34-r1-0-0
Degree $1$
Conductor $441$
Sign $0.996 + 0.0818i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.955 − 0.294i)5-s + (−0.222 − 0.974i)8-s + (0.222 − 0.974i)10-s + (0.0747 + 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.955 − 0.294i)16-s + (−0.623 + 0.781i)17-s − 19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)26-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.955 − 0.294i)5-s + (−0.222 − 0.974i)8-s + (0.222 − 0.974i)10-s + (0.0747 + 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.955 − 0.294i)16-s + (−0.623 + 0.781i)17-s − 19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (−0.623 − 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.996 + 0.0818i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.996 + 0.0818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5751156349 + 0.02357096125i\)
\(L(\frac12)\) \(\approx\) \(0.5751156349 + 0.02357096125i\)
\(L(1)\) \(\approx\) \(0.5898364632 + 0.3261177171i\)
\(L(1)\) \(\approx\) \(0.5898364632 + 0.3261177171i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.0747 + 0.997i)T \)
5 \( 1 + (-0.955 - 0.294i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−23.785108371658758754224349355803, −22.643659266025519471301049222976, −22.228056210586253143531275857505, −21.26348010789264432384897720062, −20.209373824577056794213294709585, −19.62611604804281048023283945444, −18.88020716598001399576117666377, −18.100870752517880364422172282172, −17.04368147546516741375432649069, −15.96862377236029286951873274745, −14.927775041532928265410437716446, −14.138062160146563603785766622388, −13.095482302256099940127572221410, −12.2177591671899190091341335815, −11.39717056211394582356511759366, −10.7430594534513577740293332788, −9.72705873179534211127598971205, −8.5749882747715628224091801492, −7.8875438013879166648582379972, −6.52992766945190862466155579135, −5.17776068820196358848100674887, −4.205596367451107986320130325247, −3.25102086914944363824853025509, −2.3386420098339818150732080577, −0.6702340419909561342131885897, 0.2416548913031877803093117224, 2.10745273648928902921978763378, 4.06550093032218627234179547140, 4.29299171822744805807237400408, 5.58855021600790733923887636688, 6.78817027451542585257060054096, 7.50175092222154566469937626720, 8.39020504321637277370001216071, 9.28610726957331288801694358792, 10.31719306563084494379186833184, 11.729686687934573390229057700553, 12.52874175976768245178738651593, 13.323492098895783906874672623333, 14.70041765550652520205439566944, 15.05546502369288030961553565700, 15.9657147920015060006798606850, 16.91720829613802822891260039792, 17.48193357701097392700909074997, 18.66533322320164645253198396222, 19.43994678021955684364989323546, 20.2891542193013277287999749974, 21.55475240770488839760701143571, 22.394978585500048648814109032006, 23.18898696620443816968247119730, 23.98590377482945166619282656649

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