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Properties

Label 1-21e2-441.52-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: $\chi_{441} (52, \cdot )$
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

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

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