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Properties

Label 1-3e2-9.4-r0-0-0
Degree $1$
Conductor $9$
Sign $0.766 + 0.642i$
Analytic cond. $0.0417958$
Root an. cond. $0.0417958$
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + 8-s + 10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s + 19-s + (−0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (−0.5 − 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(9\)    =    \(3^{2}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.0417958\)
Root analytic conductor: \(0.0417958\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{9} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 9,\ (0:\ ),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3545096907 + 0.1290309751i\)
\(L(\frac12)\) \(\approx\) \(0.3545096907 + 0.1290309751i\)
\(L(1)\) \(\approx\) \(0.5894898173 + 0.1730977885i\)
\(L(1)\) \(\approx\) \(0.5894898173 + 0.1730977885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
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

−47.62173401103649749753569622408, −46.08864144799801733285886812585, −45.35044946191428229582857935910, −43.35000556518745618640845067769, −41.66756822443572459347787278070, −39.60901661706606625097712505497, −38.660887703382661179546792137594, −37.142980816356091792450320751717, −35.67873662467874083280318830247, −34.17251315935910199015388133772, −31.78761076737493781801411576235, −30.23132881111232373001806532132, −29.06505615364935285125366154356, −27.05137206341616744758393680668, −26.153920713962248629788070707247, −23.29489201737135048964840603408, −21.69812885525653841996654944995, −19.76993191597980768296100382079, −18.54506475848853889876116795836, −16.50206370456441965023053675034, −13.791949509078667909084017323792, −11.58364908234498339125406731429, −10.01655042256533560545005518682, −7.532433052290406891753430961599, −3.44409315514894385547842266782, 5.31957512328119563183789417847, 7.83467143898723227716911760380, 9.62077285324898946095116559053, 12.60181339301452920379283505116, 15.12593010769490438260852250073, 16.44722843628789537974102947469, 18.32531541059901858350328038533, 20.08703614410728363126734191293, 22.72217825927669423580485115381, 24.36462870621875675984255506452, 25.64643292379770299419768200597, 27.56284572189007135431893865743, 28.639074521200676718680707660404, 31.45809895872544158373691951666, 32.5966813763290971383361066080, 34.47157502724075661222053296776, 35.60385597753093862480490580471, 36.9640446154430159417482134956, 38.8749217059289776997517862295, 40.86928099799932164741770840453, 42.22387076349533691218241379192, 43.82609081563297646192371603060, 44.63880391067545672697303237087, 46.4754515231386178212978287333, 47.80045339573958237055766271637

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