↑

Properties

Label 1-3e4-81.16-r0-0-0
Degree $1$
Conductor $81$
Sign $0.813 + 0.581i$
Analytic cond. $0.376162$
Root an. cond. $0.376162$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

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

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.813 + 0.581i$
Analytic conductor: \(0.376162\)
Root analytic conductor: \(0.376162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ 0.813 + 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443369254 + 0.4627977279i\)
\(L(\frac12)\) \(\approx\) \(1.443369254 + 0.4627977279i\)
\(L(1)\) \(\approx\) \(1.508351690 + 0.3568202175i\)
\(L(1)\) \(\approx\) \(1.508351690 + 0.3568202175i\)

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 \( 1 + (-0.286 + 0.957i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.85957254570640124719075496246, −29.86835302121381307870969230018, −29.332819340363506368172995014094, −27.66187715083165133546490908790, −26.82745339839898230028578610978, −25.41175410741605187189511429426, −24.128254881958670832080754044850, −23.2328159393720863430591556672, −22.29234775720741746147515756479, −21.38901204428253711188396114946, −19.929767870362231791586906672177, −19.38084755744708837282732320952, −17.83844439745558660579678863183, −16.42034540424357170921668621309, −14.78607536204184762790281939251, −14.41493591756224817087198250785, −13.06605282754740238731530469885, −11.58843840558957668041031612506, −10.89908417254955990918633208079, −9.59550691148332290549137474799, −7.328213203984210235769732069375, −6.51853136401614816729002635013, −4.62496920131118462907909300353, −3.6006334021083603504720132135, −1.94565726491062724902198633715, 2.26893366163448331697734330084, 4.1479505629758625210702739392, 5.17359367323644566617873763001, 6.46697841152799306771208198134, 8.10243692760131170521109028679, 9.02251325684059506861217457319, 11.254409128895010856656581688778, 12.27654696301127161276905626033, 13.08523614704048506756421331716, 14.65970630903498961339855746294, 15.40004234143337516002520447473, 16.6919008654317037232670425525, 17.53356449308408736406750324484, 19.42255620295255840852638391363, 20.459703634478849560166089033815, 21.61817018779828745279057219034, 22.40089123072816181887425961018, 23.82585130698137269410383377383, 24.63741764444074724061736902799, 25.20820111979445419856510367130, 26.84617309363004752939431632702, 27.94618815703390769701737887075, 29.09937614194138475428511884393, 30.36139110004354115832724801594, 31.29355326506814975909057241324

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