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Properties

Label 1-3e4-81.59-r1-0-0
Degree $1$
Conductor $81$
Sign $0.995 - 0.0968i$
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  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (0.835 − 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 − 0.116i)20-s + (−0.686 − 0.727i)22-s + (−0.597 + 0.802i)23-s + ⋯
L(s)  = 1  + (0.0581 − 0.998i)2-s + (−0.993 − 0.116i)4-s + (−0.973 + 0.230i)5-s + (0.597 + 0.802i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.686 − 0.727i)11-s + (0.893 + 0.448i)13-s + (0.835 − 0.549i)14-s + (0.973 + 0.230i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 − 0.116i)20-s + (−0.686 − 0.727i)22-s + (−0.597 + 0.802i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.995 - 0.0968i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.995 - 0.0968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319144983 - 0.06400410741i\)
\(L(\frac12)\) \(\approx\) \(1.319144983 - 0.06400410741i\)
\(L(1)\) \(\approx\) \(0.9544380923 - 0.2374136663i\)
\(L(1)\) \(\approx\) \(0.9544380923 - 0.2374136663i\)

Euler product

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

−30.757694830242587105301095518446, −30.20016198965782672277304166906, −27.98452329820870590298644766930, −27.577685651213247609628481554788, −26.4534444275305045454078026627, −25.365358501108140093793714190420, −24.22945546424682803117305005862, −23.30066026019023919857706307905, −22.70927955295718166831115809419, −20.98053577687851797659776638049, −19.83336377940792012896691415748, −18.52573520218044133047923691726, −17.316334251832771652436395209136, −16.41247388598860203728242110934, −15.246536043473930778475365182405, −14.34617002477669217570995121341, −13.03226883722862622200547842025, −11.7256011957965632251892732211, −10.150391582934874880509918135761, −8.509112839843962053942580000126, −7.69876900724556852403759875103, −6.47651925252235632222972295379, −4.706849908844874930677044655247, −3.847220132739422287990209419835, −0.7567806208400374489783704841, 1.366477733955942754005466486664, 3.19330514990299328748306998589, 4.31868310749863679989037567318, 5.96901027668593055310559192606, 8.10961565670858111002636243608, 8.91620371384591359013336783182, 10.64497134596835764517819544867, 11.6092690874616166026455620283, 12.33808469240489253974958697323, 13.97984759220641738599748054075, 14.939171205949179704444809824233, 16.41004124507860243654560774771, 17.99570393082350223099580951004, 18.95725488623569863056239213933, 19.66997022643201349901002664936, 21.12505017034725994572243121132, 21.79357807785331076025281675740, 23.164474090260443314369269159020, 23.88709871907000009727868641357, 25.50032449050005935926447942779, 26.963864995462390272882233914203, 27.64977987975326564451881933145, 28.43681281070633870635319087725, 29.91598671227780470617137708450, 30.60862160010000405265784908855

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