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Properties

Label 1-21e2-441.13-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: $\chi_{441} (13, \cdot )$
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

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

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