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Properties

Label 1-21e2-441.193-r0-0-0
Degree $1$
Conductor $441$
Sign $0.566 - 0.824i$
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  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)8-s + (0.365 − 0.930i)10-s + (−0.222 + 0.974i)11-s + (0.955 − 0.294i)13-s + (0.365 − 0.930i)16-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.826 − 0.563i)4-s + (0.623 − 0.781i)5-s + (0.623 − 0.781i)8-s + (0.365 − 0.930i)10-s + (−0.222 + 0.974i)11-s + (0.955 − 0.294i)13-s + (0.365 − 0.930i)16-s + (0.826 + 0.563i)17-s + (−0.5 + 0.866i)19-s + (0.0747 − 0.997i)20-s + (0.0747 + 0.997i)22-s + (−0.900 − 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.566 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.566 - 0.824i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.566 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.435736494 - 1.281779910i\)
\(L(\frac12)\) \(\approx\) \(2.435736494 - 1.281779910i\)
\(L(1)\) \(\approx\) \(1.945444676 - 0.6404635519i\)
\(L(1)\) \(\approx\) \(1.945444676 - 0.6404635519i\)

Euler product

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

−23.858147601758776383058020121079, −23.52300632599348395964124694620, −22.302525740771447820877786708519, −21.808896973733511923563242146, −21.06844334734775767933196187274, −20.18143582545868419359092988528, −18.9231656956839972398026120632, −18.20485619942940767512273129645, −17.05209475291492750141875535124, −16.246202025632082740965762822250, −15.37059293273086385663423100749, −14.4225589882219772191886246971, −13.68081060186303023136658321042, −13.17054998619210637817041001610, −11.76693727269360239876167411114, −11.10431872456713309755941488968, −10.200221380182445179999631839104, −8.828112318677146522455649305699, −7.72812939201157362044979910439, −6.65815221253488852721867186931, −5.97953971592208093683680278760, −5.080569772314174083715160602279, −3.64738082869335446687561546287, −2.95911126068859692951132927719, −1.722258357019340128840675486766, 1.36685357089667312451222835357, 2.19106663356984632725328461880, 3.65831440058393630699949011229, 4.525366212101637923553277445822, 5.62538303282321651672083804739, 6.1990609955215248840304773121, 7.574662244464691334044636023677, 8.67938018817957648004073527766, 10.048878545330943489635123503265, 10.451209639340640268416333277991, 11.943995775988861390538223864709, 12.531850880285912303167421509273, 13.298664000374859842852625845300, 14.16705102037511389363619189826, 15.05613579066261699964953410171, 16.03184908876396182132827621553, 16.79417314775277715687996061262, 17.88848325027073883488755590296, 18.89026467136547606387556895859, 20.069732034653801125290915887529, 20.632925667011896733074654592793, 21.28418341078073748745557468423, 22.1306812174518032726965417290, 23.352384925743899405440252292359, 23.496806842685084040906635947964

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