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Properties

Label 1-21e2-441.164-r0-0-0
Degree $1$
Conductor $441$
Sign $0.944 - 0.328i$
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  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (0.900 − 0.433i)8-s + (−0.0747 − 0.997i)10-s + (0.988 − 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.900 − 0.433i)16-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (0.900 − 0.433i)8-s + (−0.0747 − 0.997i)10-s + (0.988 − 0.149i)11-s + (0.988 − 0.149i)13-s + (−0.900 − 0.433i)16-s + (−0.733 + 0.680i)17-s + (0.5 − 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.955 + 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 − 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184690693 - 0.2002018180i\)
\(L(\frac12)\) \(\approx\) \(1.184690693 - 0.2002018180i\)
\(L(1)\) \(\approx\) \(0.9581280186 - 0.1811702982i\)
\(L(1)\) \(\approx\) \(0.9581280186 - 0.1811702982i\)

Euler product

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

−24.34599687028608164981909248462, −23.47503246849409929173297971314, −22.49287985718909121788123542476, −21.64783919650259758443786122449, −20.28362368135997558045028089568, −20.010301550146678007671739148617, −18.48396736169574569302129896668, −18.1044541806381248559570606817, −17.09314665949835313577282773392, −16.39379101435418096857915420237, −15.70819615454502224339832930933, −14.32860104402620969220959513229, −13.964985728863461006903182269384, −12.85757452125500404101352389315, −11.61063190695128421762732770336, −10.50450853588852800707488338405, −9.51560446093543050839074255323, −8.94861792562027856216542727864, −8.01983928270391738365956645792, −6.72030117720786981595478573816, −6.076054994087077186287009521951, −5.084888113358242524854438846351, −3.960482142070331787616013903230, −2.028208983174518305674190404248, −1.077451289424739513236490555150, 1.21623232814208761595106020230, 2.22297468412340634347398407897, 3.34125615567759615552730703775, 4.320430743401116635612226065158, 5.958097565918597240588922910131, 6.77710558070179447246565010587, 8.01228214476676313252654489508, 9.07780482249313761973149812897, 9.66722735396660912614566392005, 10.82890052397981680961257054652, 11.28098898062376111060973637815, 12.4734480225644885062348815617, 13.4867428853466223615542043958, 14.05958683227866707213299069604, 15.39740096357456372934940594388, 16.45698817383928428510092953169, 17.51465006861350781271702156891, 17.885871340744666326724890837816, 18.84644070792921379480796807810, 19.72738634236275977505213154597, 20.46703059167463229297496631500, 21.56457178220403321383971977345, 21.99792355517781029583060264819, 22.787444431435158118209638335805, 24.105659453761505652904013058571

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