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Properties

Label 1-21e2-441.319-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.995 + 0.0924i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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 + (−0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)10-s + (−0.900 − 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (−0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)10-s + (−0.900 − 0.433i)11-s + (0.0747 + 0.997i)13-s + (0.955 − 0.294i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.623 + 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.995 + 0.0924i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.995 + 0.0924i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02220149874 + 0.4791230940i\)
\(L(\frac12)\) \(\approx\) \(0.02220149874 + 0.4791230940i\)
\(L(1)\) \(\approx\) \(0.6352601638 + 0.3479833751i\)
\(L(1)\) \(\approx\) \(0.6352601638 + 0.3479833751i\)

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 \( 1 + (-0.222 - 0.974i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.365 + 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (-0.733 + 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
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 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.0747 - 0.997i)T \)
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

−23.210890768554326849901464110988, −22.76336766072263211997083972750, −21.903839371421132468691706830, −21.10620693611478238046540732511, −20.10964710915905962857783039615, −19.50554115853759177221445766666, −18.38729817851699329293606127578, −18.04359325523719108876987468230, −17.02864062871267778249041052018, −15.235707461698262489730923619593, −15.15835922611688126892136081263, −13.699879872798752122787004788627, −13.07884327499510867985545733579, −12.12072891523685845649016451484, −10.85274886673994240921054811802, −10.72225914675723863787731040317, −9.59699202980649102346982363488, −8.44430525941540254267697483457, −7.487952916684733547970137329665, −6.240773719447567908005691966431, −5.01719555984627267051544280049, −4.01240634085856150813916654592, −2.80983253293363086119645628542, −2.238721232648869093804936641, −0.26509121500627026694837282435, 1.50543217125878729901125982551, 3.38227780388761943161766590514, 4.54798421449295643177511107715, 5.19514987567947238612337124606, 6.30615584562975757380009684319, 7.318435113420956805185075546797, 8.3960101589456560571931044306, 8.88444819815612031870055919536, 9.96166292211662747135373919432, 11.28683026800966070939201365713, 12.45631484348906147398451679648, 13.219430826540466906293344095265, 13.94797804111897459796967519108, 15.08640454042465540615835291303, 15.94569629626643314583615768790, 16.508314078485148237583091615547, 17.31296950241031007277866653084, 18.33821733878130665927295796991, 19.11952005582209891762273055124, 20.20760573911403267800520524773, 21.29679505392871271378905492076, 21.84645203035102901843864653680, 23.2555675054304167653501085808, 23.65564566462580235096646084118, 24.38593715858072724864262958895

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