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Properties

Label 1-21e2-441.304-r1-0-0
Degree $1$
Conductor $441$
Sign $0.902 + 0.430i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)10-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.365 + 0.930i)5-s + (0.623 + 0.781i)8-s + (0.988 + 0.149i)10-s + (0.955 + 0.294i)11-s + (−0.955 − 0.294i)13-s + (0.623 − 0.781i)16-s + (−0.0747 − 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 − 0.997i)20-s + (0.0747 − 0.997i)22-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171668392 + 0.2652311678i\)
\(L(\frac12)\) \(\approx\) \(1.171668392 + 0.2652311678i\)
\(L(1)\) \(\approx\) \(0.8236059955 - 0.1628446331i\)
\(L(1)\) \(\approx\) \(0.8236059955 - 0.1628446331i\)

Euler product

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

−23.96461061487898205639431849043, −23.179836190937179201918432104461, −22.235389496944874832564181996737, −21.38947642053249823860705078744, −20.09899592333687146412184806162, −19.41169460158783978670341562174, −18.60718593637506908419159658693, −17.301114231755706345154505068141, −16.84162647429881282073925886008, −16.19143033615863067745957480615, −15.01266824074375237490755559265, −14.477273785943353288975766127606, −13.30494321238271905540826360225, −12.49386947610494740109484644154, −11.506880299354229251044766461495, −10.05864804520378847306437730415, −9.23781248174903600251493686795, −8.42983382485464657483874834838, −7.60194344953457226660085390422, −6.52164619131666442118702078675, −5.54051049096106545237790559602, −4.54245811146534952473550626934, −3.721981631443335156594945920698, −1.62948127710015633980376514849, −0.45638089052305783828282170208, 0.93447387796304048642602520284, 2.41663305939565949873871710803, 3.17686713387171371043163560207, 4.26190741564497378525294470132, 5.33066422604433183204051504826, 7.01870449995093988655105332233, 7.53126259548067481998839059797, 9.0462948948347522820608238457, 9.62537058143164508506342060728, 10.74414555537481321958903361517, 11.45805636656465205234585624973, 12.172712915407320799054337391466, 13.242808205551092002255686594959, 14.304240138383523507235625883829, 14.88097226799619118483598258039, 16.192510068166016385448211528153, 17.349218605992555310908040665878, 17.99592273968678874680437525296, 18.84709119798040974405938330051, 19.80358216927837411510689707236, 20.08985644454829272027706854244, 21.49978878497838326308564610877, 22.15653250163710352689834415430, 22.71660520920660422852006376985, 23.60881432996894840881390548817

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