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Properties

Label 1-21e2-441.241-r1-0-0
Degree $1$
Conductor $441$
Sign $0.0142 + 0.999i$
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  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (−0.900 − 0.433i)8-s + (−0.826 − 0.563i)10-s + (0.365 + 0.930i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (−0.900 − 0.433i)8-s + (−0.826 − 0.563i)10-s + (0.365 + 0.930i)11-s + (−0.365 − 0.930i)13-s + (−0.900 + 0.433i)16-s + (−0.955 + 0.294i)17-s + (0.5 − 0.866i)19-s + (−0.955 + 0.294i)20-s + (0.955 + 0.294i)22-s + (−0.733 + 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0142 + 0.999i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0142 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3210523905 - 0.3165104481i\)
\(L(\frac12)\) \(\approx\) \(-0.3210523905 - 0.3165104481i\)
\(L(1)\) \(\approx\) \(0.8145093148 - 0.7041521116i\)
\(L(1)\) \(\approx\) \(0.8145093148 - 0.7041521116i\)

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 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (-0.365 - 0.930i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.733 + 0.680i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.733 - 0.680i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.826 + 0.563i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
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 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.365 + 0.930i)T \)
89 \( 1 + (0.988 - 0.149i)T \)
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

−24.304728180478562368243568774809, −23.79085840084341432892629671072, −22.633050480639708752691051006739, −22.14347608511706288459921386151, −21.47688660921594660585583017000, −20.34783818132380124178825059722, −19.13688493659942330609898654390, −18.368016878531860871009393755724, −17.495568865050372992365953250678, −16.425742626900071423769012930682, −15.85163621337970645882259941196, −14.68665734589167805201792713102, −14.15728437147079850745427414010, −13.471819770923624533048362719181, −12.089795284447404960405755528453, −11.49726071787989743574851221357, −10.33862208339452111688067685771, −9.05216309894299003812562160051, −8.151221313121151853413815360699, −6.977630049904872825715232443731, −6.49116208204444650890458084676, −5.41970070632619522548038906537, −4.14821517584917094036545879817, −3.338690304512959456268849819167, −2.17248199143566483257828996248, 0.09254913615991579111958936263, 1.34475495523772623791634895635, 2.395041148427689995462073110639, 3.74483492896558419485410378664, 4.69268332400747895253313904416, 5.3727808023160505054091730997, 6.60512628470980654790238919806, 7.92932182774040478847769748028, 9.16426000797151702479547132240, 9.7670681582773779075922050977, 10.9045686752230443376517027521, 11.885753025221566676080186153024, 12.63403597014752361910550310224, 13.25012589937444286792305727117, 14.27119272092809838674616360174, 15.35681684610046546048546310851, 15.91430196453522609524863178863, 17.50863574445697163839434657990, 17.81745277706057440920072754001, 19.37030060549756699163509298773, 20.02733631939986382557705202320, 20.399900917818234677280175673582, 21.54784196283550466151448799442, 22.232828781276700866567183771277, 23.11718969690054768233117715318

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