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Properties

Label 1-3e4-81.47-r1-0-0
Degree $1$
Conductor $81$
Sign $0.0774 + 0.996i$
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  + (0.686 + 0.727i)2-s + (−0.0581 + 0.998i)4-s + (0.993 − 0.116i)5-s + (0.893 + 0.448i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.396 − 0.918i)11-s + (0.973 + 0.230i)13-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (0.0581 + 0.998i)20-s + (0.396 − 0.918i)22-s + (−0.893 + 0.448i)23-s + ⋯
L(s)  = 1  + (0.686 + 0.727i)2-s + (−0.0581 + 0.998i)4-s + (0.993 − 0.116i)5-s + (0.893 + 0.448i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (−0.396 − 0.918i)11-s + (0.973 + 0.230i)13-s + (0.286 + 0.957i)14-s + (−0.993 − 0.116i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (0.0581 + 0.998i)20-s + (0.396 − 0.918i)22-s + (−0.893 + 0.448i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0774 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $0.0774 + 0.996i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.0774 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.107106244 + 1.949684296i\)
\(L(\frac12)\) \(\approx\) \(2.107106244 + 1.949684296i\)
\(L(1)\) \(\approx\) \(1.618427994 + 0.9056868716i\)
\(L(1)\) \(\approx\) \(1.618427994 + 0.9056868716i\)

Euler product

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

−30.44301539641359780216486050780, −29.62714388633109677570489635641, −28.512342501887479683407637086174, −27.68537814799999714097686393658, −26.18211315912644746651723984044, −24.95458977886989231078564168612, −23.85572072699129567125402998304, −22.82340360804161002348031078821, −21.71762215473169495203786614091, −20.72653220080552851140016662160, −20.11469131127612700402930019846, −18.23374502131793768508457348818, −17.79816513339196796030092888669, −15.88114582972065334208771313046, −14.48477318180073022756282988572, −13.728384886403399584268737648928, −12.65556419790930653522462179869, −11.171591978196345202538129213205, −10.3199914924519273068502816932, −9.04197289101916887936745111176, −7.036338781840297695364750657682, −5.5262019200808615016911638096, −4.47143921374918441994420529926, −2.637960689681680118548723315457, −1.323282850427507990554958103325, 2.00078796799759549938461208327, 3.84046721260376598415495361839, 5.51545124185131524715381398673, 6.09773321236418388722299949259, 7.97154210078463746095180764839, 8.88633722057483845053778655391, 10.75047072932560272992738109965, 12.12821747048661924804010665683, 13.48535280733684881744985454359, 14.153261517630565279595445187603, 15.4207386499413830148193487753, 16.60678442544208042213153238469, 17.65867984131688942140664279913, 18.60436494297736017426025327176, 20.79258528578100210335787438524, 21.31834705108973595129495204887, 22.28562382980713243681153443712, 23.77213635188537395279032600420, 24.444520485745381868306379402464, 25.514448479569282810459371923331, 26.36170610024523076994022417828, 27.7252178152504595715873396119, 29.06305562607839197476821128007, 30.16010975498660174662513943913, 31.10516220359868132228959230959

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