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Properties

Label 1-3e4-81.5-r1-0-0
Degree $1$
Conductor $81$
Sign $-0.581 - 0.813i$
Analytic cond. $8.70465$
Root an. cond. $8.70465$
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  + (−0.893 + 0.448i)2-s + (0.597 − 0.802i)4-s + (0.286 − 0.957i)5-s + (0.396 − 0.918i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.973 − 0.230i)11-s + (−0.835 + 0.549i)13-s + (0.0581 + 0.998i)14-s + (−0.286 − 0.957i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.597 − 0.802i)20-s + (0.973 − 0.230i)22-s + (−0.396 − 0.918i)23-s + ⋯
L(s)  = 1  + (−0.893 + 0.448i)2-s + (0.597 − 0.802i)4-s + (0.286 − 0.957i)5-s + (0.396 − 0.918i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (−0.973 − 0.230i)11-s + (−0.835 + 0.549i)13-s + (0.0581 + 0.998i)14-s + (−0.286 − 0.957i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.597 − 0.802i)20-s + (0.973 − 0.230i)22-s + (−0.396 − 0.918i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.581 - 0.813i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ -0.581 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2943714876 - 0.5722389391i\)
\(L(\frac12)\) \(\approx\) \(0.2943714876 - 0.5722389391i\)
\(L(1)\) \(\approx\) \(0.6252418340 - 0.1590045646i\)
\(L(1)\) \(\approx\) \(0.6252418340 - 0.1590045646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.893 + 0.448i)T \)
5 \( 1 + (0.286 - 0.957i)T \)
7 \( 1 + (0.396 - 0.918i)T \)
11 \( 1 + (-0.973 - 0.230i)T \)
13 \( 1 + (-0.835 + 0.549i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.396 - 0.918i)T \)
29 \( 1 + (0.0581 - 0.998i)T \)
31 \( 1 + (-0.993 - 0.116i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.893 - 0.448i)T \)
43 \( 1 + (-0.686 + 0.727i)T \)
47 \( 1 + (0.993 - 0.116i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.973 + 0.230i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (-0.893 + 0.448i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.286 - 0.957i)T \)
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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

−30.851386484322309000471364272151, −29.84902327988469487321507677625, −29.02879909483027524611458743688, −27.79796739434868899222903450370, −27.040428612652859484398072898971, −25.717391275924700721609423621903, −25.27195279431551182581711282701, −23.641997686286220823444532872021, −22.009523316760893387055911829126, −21.451719073328169945505927399734, −20.128333608902480064289688133952, −18.79864653841362485324523560823, −18.22428711711006227825038481895, −17.20526739270682374659975252865, −15.64157308430505904690158194864, −14.71558216150489608501368412258, −12.914076205655575175036423796062, −11.73970095406688931223384541186, −10.57588089487596808117121760751, −9.65300584182294955477748365550, −8.17028679423323688195563145803, −7.10901284861331844965300726098, −5.45407638974052896638911793730, −3.096617172885854722053188360556, −2.0802693532948631044719919276, 0.38632953336235967432726825018, 1.97163382437959541489775673631, 4.56209655633493851588308370735, 5.878995750132867493563806061036, 7.50768636320174443487761856854, 8.39624904373662494739136015968, 9.77997679705889431461531773990, 10.710087142449619672726673602066, 12.30511297087542320233750378267, 13.77931062745880066719479157495, 14.971053940012342628798220131, 16.581420856404359545057294979614, 16.85495111263874114706927363318, 18.16808314634428468661153430705, 19.40278547284616877188519419175, 20.48139964339661762473508761244, 21.29446079363915736186468340494, 23.44073831568712240097435586132, 23.992992913874973910573512535827, 25.077971129645896155096321053460, 26.2152857129295422291071553453, 27.097330897586450056114246983, 28.18114422398738147895691098562, 29.07090643987147045323368382167, 30.00299828116942826687522153525

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