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Properties

Label 1-3e4-81.11-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: $\chi_{81} (11, \cdot )$
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

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

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