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Properties

Label 1-21e2-441.83-r0-0-0
Degree $1$
Conductor $441$
Sign $0.0818 - 0.996i$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (0.222 − 0.974i)10-s + (−0.0747 − 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s − 19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (0.222 − 0.974i)10-s + (−0.0747 − 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s − 19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0818 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0818 - 0.996i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.0818 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9745105110 - 0.8977753723i\)
\(L(\frac12)\) \(\approx\) \(0.9745105110 - 0.8977753723i\)
\(L(1)\) \(\approx\) \(0.9586565136 - 0.5177897445i\)
\(L(1)\) \(\approx\) \(0.9586565136 - 0.5177897445i\)

Euler product

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

−24.45145559237454721158504607538, −23.43644727090785183874330255645, −22.7817033771335850253645140277, −21.729696126321687342438345877898, −21.116555933649891840526498450212, −19.84551825153016117838745536190, −18.93852971442337315197326155504, −17.7846090838294355803907125094, −17.339489765918176820201874838469, −16.68333703254517732719997520100, −15.431528841346110536095631318752, −14.77344981357862253800860010922, −13.96185257408777754683938713437, −12.81979500059493996905163665282, −12.468302629956864590270124345867, −10.48054206831600950124554090158, −9.90720069158956722903517741401, −8.96588012041062234885549071662, −8.00914964511891710906413901203, −6.97776498653670807383957726498, −6.08750999023126057456667976537, −5.12549261407360602175889264663, −4.37109230736715380485196547258, −2.68811156765797684059904275652, −1.2414291239009771206501975805, 0.933668113512090970256146832994, 2.3318492464256377548170544982, 2.9692746296514318283772319093, 4.37952311041300951348849565104, 5.37733468289215918956195726393, 6.428737594361278452917964693196, 7.79074078026266756266475279989, 9.00610708360144984379085214756, 9.60348324155772109395852336438, 10.60585971076257417435984421865, 11.29656822731302426719055332524, 12.40187261819016872779763761193, 13.24125159162514049680788096840, 14.118360679179862177857076576678, 14.6618999589997547957512609061, 16.34169331522576448376030773158, 17.167047606764403676908180630987, 17.90515069038013701828775173821, 19.03260662870255409163248597557, 19.23894454826923081074200483439, 20.72559884278293418914219140103, 21.23113272809695541634824750710, 21.91546070011824824273339460882, 22.72088239618027073389294058030, 23.691701812711350321385291056969

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