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Properties

Label 1-21e2-441.151-r0-0-0
Degree $1$
Conductor $441$
Sign $0.999 + 0.0142i$
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  + (0.623 − 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

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

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0142i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.999 + 0.0142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.763050726 + 0.01255982064i\)
\(L(\frac12)\) \(\approx\) \(1.763050726 + 0.01255982064i\)
\(L(1)\) \(\approx\) \(1.402127059 - 0.2337968930i\)
\(L(1)\) \(\approx\) \(1.402127059 - 0.2337968930i\)

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

−24.08545529561448210590195739220, −23.41599919353981526915848761128, −22.43224939090173016302279086649, −21.54338863372900943136463773605, −20.88060909703072462611968722860, −19.96121076267181356107701426612, −18.84702510316327646688759291791, −17.57595312840318394388998062789, −17.0691965211102888224458448870, −16.07179958429122706624367634513, −15.58702021048039918824603763338, −14.31777810953604733775509678498, −13.65113669096387161255277297702, −12.68990629341580197927209981005, −12.11856257033468971290161555951, −10.882433902392213112769386611242, −9.51610414140041925369516074490, −8.36859673142948311341274091750, −8.10256714622347595529377541135, −6.55021482230007848301982085327, −5.77504579271788527729017151917, −4.87750462611653183498224403220, −3.88352246882320435752330082068, −2.78057358490670704618918650040, −0.89059309411630179795209905297, 1.53267662820050659911587739028, 2.47387599130317080414845539216, 3.64126941478483919998373169605, 4.408467299615358422468656249873, 5.801557282449649313529968889517, 6.56312372125647132661760099672, 7.66634867500010107773565854469, 9.2260300372887915536216069633, 10.03890333005711469056943552940, 10.75016912172837341235651659030, 11.88831017573715533897749712674, 12.298032383196349926069506264758, 13.80792968691292152749333137088, 14.17594908737485451004771464801, 15.06689302354470371472517373125, 15.97720623746494788983026124223, 17.39320409076682876894733844521, 18.27133798842441265407666363104, 19.074372433369839032792984827375, 19.67118646166382240955226456397, 20.94833298002359875710514322292, 21.34698221123086510724161343594, 22.4400325690685418600535951874, 23.035119589065479440354198213311, 23.628041676406981732861059580866

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