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Properties

Label 1-21e2-441.355-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: $\chi_{441} (355, \cdot )$
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

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

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