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Properties

Label 1-3e4-81.32-r1-0-0
Degree $1$
Conductor $81$
Sign $-0.413 + 0.910i$
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  + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.686 + 0.727i)5-s + (−0.993 + 0.116i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.686 + 0.727i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.396 + 0.918i)20-s + (−0.286 + 0.957i)22-s + (0.993 + 0.116i)23-s + ⋯
L(s)  = 1  + (0.835 + 0.549i)2-s + (0.396 + 0.918i)4-s + (0.686 + 0.727i)5-s + (−0.993 + 0.116i)7-s + (−0.173 + 0.984i)8-s + (0.173 + 0.984i)10-s + (0.286 + 0.957i)11-s + (−0.0581 − 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.686 + 0.727i)16-s + (0.939 + 0.342i)17-s + (−0.939 + 0.342i)19-s + (−0.396 + 0.918i)20-s + (−0.286 + 0.957i)22-s + (0.993 + 0.116i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.413 + 0.910i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ -0.413 + 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.402576752 + 2.178228007i\)
\(L(\frac12)\) \(\approx\) \(1.402576752 + 2.178228007i\)
\(L(1)\) \(\approx\) \(1.424728376 + 0.9864647860i\)
\(L(1)\) \(\approx\) \(1.424728376 + 0.9864647860i\)

Euler product

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

−30.26174555158198949702583090196, −29.25814586698992648292809329498, −28.77244550738361071484568403848, −27.52432808924912617844685021482, −25.902341354888534876536073894689, −24.83376230700622752125413089526, −23.861070545368515419228886582169, −22.76260815511828960162373702475, −21.565989914840699157463571722949, −20.9704647138325722658982135582, −19.56188095223965592853675707073, −18.84110826473488276677965157952, −16.87782043005642060893956361671, −16.09421629229657639692170988042, −14.464279439936236135852200825101, −13.44780354925966402474786135781, −12.64895737319117713765552263910, −11.3649914832089562019257652482, −9.94121418748831477506142600099, −8.98949800879504353476493210319, −6.666624887796329505546331749075, −5.669428276333028165983680818479, −4.22834676465989006836272105036, −2.74788646879065121450699466502, −0.99905240213455685257163105992, 2.46008071686450395985394139731, 3.7023427178147797484536959100, 5.51512129603439117369194469764, 6.48899332106191463483099980223, 7.61091702086558433545472806666, 9.46321995674256930061274779133, 10.727187826967977314706697634032, 12.46939411840985545705908473809, 13.15595313677111674719219945772, 14.59508075696443180101413977483, 15.25237744646812833477885277429, 16.72193099233670147002327997032, 17.637167479907422425548431976581, 19.065137916741682596572678061878, 20.52893350575259303513326889263, 21.64707829811456804742938736013, 22.69537818074073384689811433357, 23.14671716916319761612973969318, 24.95003823362565611279097854038, 25.52167360050350290841611288498, 26.3167539868539417738444804048, 27.90272768680826520098807350356, 29.48766994442803375981870778627, 29.96172059860744285815638646448, 31.16751736275198203483628185985

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