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Properties

Label 1-21e2-441.229-r1-0-0
Degree $1$
Conductor $441$
Sign $0.968 - 0.250i$
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  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.988 − 0.149i)5-s + (0.623 + 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.733 + 0.680i)11-s + (0.733 − 0.680i)13-s + (0.623 − 0.781i)16-s + (−0.826 + 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.826 + 0.563i)20-s + (0.826 + 0.563i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.988 − 0.149i)5-s + (0.623 + 0.781i)8-s + (−0.365 − 0.930i)10-s + (−0.733 + 0.680i)11-s + (0.733 − 0.680i)13-s + (0.623 − 0.781i)16-s + (−0.826 + 0.563i)17-s + (0.5 + 0.866i)19-s + (−0.826 + 0.563i)20-s + (0.826 + 0.563i)22-s + (0.0747 − 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.755687113 - 0.2231927434i\)
\(L(\frac12)\) \(\approx\) \(1.755687113 - 0.2231927434i\)
\(L(1)\) \(\approx\) \(1.002533130 - 0.3316711080i\)
\(L(1)\) \(\approx\) \(1.002533130 - 0.3316711080i\)

Euler product

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

−23.9279964549720975059451509353, −23.33210894580542384651225787235, −22.04140191405227619265230272298, −21.679168179086280592115151610871, −20.50871291212376612849000051541, −19.30509349225239773613888699524, −18.31526870772160580819526949325, −17.90572419248504110230393732151, −16.932429678751971783887813334265, −16.01446768728608952610395042699, −15.412126006454060408052293207363, −13.97116705254271469293043279959, −13.7815934122641632667025938875, −12.860359562809687556874160327150, −11.21209534456926637024003863878, −10.40829972387837019533976214590, −9.18376259892271563472002332531, −8.83780426412211531655774266313, −7.42324835450537612653085659005, −6.63568783219007908041560479619, −5.65221647346776859629009184745, −4.9608717243941645734156478941, −3.508299677921043321143228378088, −2.026845435938753335128635327433, −0.60476961966348004035105191481, 1.01786811403074776390801649682, 2.07106940182737368885983541996, 2.99383882659285342359769547417, 4.3338711385689642313810842906, 5.2994038988309327206007784454, 6.3783942723791273346520588451, 7.91783504054164038649539550983, 8.6872043603830452772036234608, 9.81863438619658340509265198186, 10.34355295173222776618420436379, 11.2347633512961539965381880306, 12.57670987017753521241785306768, 13.00581373702725408750169642317, 13.88564449651834787069781577445, 14.89882109843943425227826534381, 16.19225179761223313372357763952, 17.197426290855261559627066319777, 18.066588751148921884530411312773, 18.38322480196774808823615700318, 19.687727116640741543825681579303, 20.63131363176481889508401119788, 20.923397356149041655748176365157, 22.0285622548189590585505526476, 22.66947436624222463638383194443, 23.59938046345374053148059428405

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