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Properties

Label 1-21e2-441.74-r1-0-0
Degree $1$
Conductor $441$
Sign $0.427 + 0.904i$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.733 − 0.680i)5-s + (0.222 + 0.974i)8-s + (0.955 − 0.294i)10-s + (−0.826 + 0.563i)11-s + (0.826 − 0.563i)13-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.733 − 0.680i)5-s + (0.222 + 0.974i)8-s + (0.955 − 0.294i)10-s + (−0.826 + 0.563i)11-s + (0.826 − 0.563i)13-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.988 + 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.988 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.427 + 0.904i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.427 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.631409660 + 2.299847964i\)
\(L(\frac12)\) \(\approx\) \(3.631409660 + 2.299847964i\)
\(L(1)\) \(\approx\) \(2.038331080 + 0.6687461703i\)
\(L(1)\) \(\approx\) \(2.038331080 + 0.6687461703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (-0.826 + 0.563i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.365 + 0.930i)T \)
29 \( 1 + (0.988 + 0.149i)T \)
31 \( 1 + T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.365 + 0.930i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.826 - 0.563i)T \)
89 \( 1 + (-0.0747 + 0.997i)T \)
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

−23.484259202234620613540734445955, −22.8550143827001945048554781751, −21.8170613765653941544196529478, −21.24884001602337835119623605522, −20.66376202016356801163225548394, −19.32527262860579110613211919673, −18.71783566065949251974660611239, −17.86518549503421384340680127509, −16.518357608406125722427388359735, −15.71238710022250883383801820231, −14.70326642499265663622559716878, −13.8843744719700626480180913345, −13.36749377751820016576986325379, −12.30284830062513406167280736196, −11.210808395396358631463917964851, −10.5547607882294310142877488235, −9.75404289783329670952222879573, −8.421724667735457537749582018514, −6.991928416997005878581975448978, −6.19065206646571516587744840923, −5.390835361213615701022920202817, −4.20265202235596637857807977947, −2.99777233189454788066192316229, −2.29766809307763985333449220759, −0.88879707166844515991865588147, 1.310734689344574316628652449060, 2.526209921493490769821780410741, 3.71249081669771973458966937190, 4.87532095597878589295173717444, 5.637275717857427013975547295818, 6.40693672603248807013005864694, 7.81853574243361935513986728102, 8.38274636887347708620601987601, 9.83593887396524700943077907754, 10.66878700421535254238687572205, 12.08411215450486094338192088210, 12.65378511163977737186892983471, 13.53876587997145143395216591062, 14.18213484786842272363256665119, 15.395450010229176462264653909818, 15.976119575829430439037765485285, 17.02367446159321394106908902648, 17.630073281198908964714156629725, 18.73548680603919986813037455012, 20.17850707108729468063558958997, 20.822300417378716972285126707910, 21.3393702045017531919845345686, 22.33562220456849725693223597261, 23.45035007592645891363657760375, 23.66036832611995857572414825726

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