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Properties

Label 1-21e2-441.283-r1-0-0
Degree $1$
Conductor $441$
Sign $-0.569 - 0.822i$
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  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)5-s + (−0.222 + 0.974i)8-s + (0.733 − 0.680i)10-s + (−0.900 + 0.433i)11-s + (−0.826 − 0.563i)13-s + (−0.733 + 0.680i)16-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)5-s + (−0.222 + 0.974i)8-s + (0.733 − 0.680i)10-s + (−0.900 + 0.433i)11-s + (−0.826 − 0.563i)13-s + (−0.733 + 0.680i)16-s + (−0.365 + 0.930i)17-s + (0.5 − 0.866i)19-s + (0.988 − 0.149i)20-s + (−0.988 − 0.149i)22-s + (0.623 − 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.569 - 0.822i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ -0.569 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2525363503 - 0.4819698258i\)
\(L(\frac12)\) \(\approx\) \(0.2525363503 - 0.4819698258i\)
\(L(1)\) \(\approx\) \(1.233605591 + 0.2262104711i\)
\(L(1)\) \(\approx\) \(1.233605591 + 0.2262104711i\)

Euler product

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

−24.02067593774899847479582666910, −22.98515082091156429636784934998, −22.503622272481080667655349944230, −21.54116736527548539882705681449, −20.99249443904316598423775197433, −19.94328743850227137509700472622, −18.89832924253418628378469021675, −18.54236402580316789580277595706, −17.32166235374382975710749711015, −15.969453750559743257858004590662, −15.31071734702288783994432071689, −14.18860347202591458415758846649, −13.84421966351064273730848886682, −12.72948282035479184134437150420, −11.702604719757963315533792491708, −10.98954505816914884822396324982, −10.1106961204418101404871567082, −9.31191691854827888250544055578, −7.578536916105787007428978141292, −6.81014780583062860355007898138, −5.67879062385533511175247201361, −4.87974093945803561953276905399, −3.48369304332304796256037505239, −2.759680749060287823173630265378, −1.70226551037779105085907451860, 0.0940170604697869914681573697, 1.956818435111577734676282503378, 3.061529367338502278364282668336, 4.50331430665923522274403766890, 5.04900497394272682361703496997, 5.98607385188758733993243840099, 7.214843181354015864596729788892, 8.0553987877626512897329624644, 8.97211481350359495348573886532, 10.17653056585493834805263134336, 11.37884993653575483082592375855, 12.54977704836153146029019752909, 12.91570326837711870900648120541, 13.76196212195571263415064297263, 15.0404890348024638298396987559, 15.48760354494901092494137270771, 16.59703281651582143331186426807, 17.24562322484006270800672922905, 18.01544924041171084195207169522, 19.51304506634007180837923420965, 20.46461252953047396338329545240, 20.96035628865376667006518989544, 21.99126580428474582073340483491, 22.6740466605430598640346921684, 23.80184521981421939825987647017

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