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Properties

Label 1-3e4-81.20-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  + (−0.973 + 0.230i)2-s + (0.893 − 0.448i)4-s + (−0.597 − 0.802i)5-s + (−0.835 + 0.549i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (0.993 + 0.116i)11-s + (−0.286 − 0.957i)13-s + (0.686 − 0.727i)14-s + (0.597 − 0.802i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.893 − 0.448i)20-s + (−0.993 + 0.116i)22-s + (0.835 + 0.549i)23-s + ⋯
L(s)  = 1  + (−0.973 + 0.230i)2-s + (0.893 − 0.448i)4-s + (−0.597 − 0.802i)5-s + (−0.835 + 0.549i)7-s + (−0.766 + 0.642i)8-s + (0.766 + 0.642i)10-s + (0.993 + 0.116i)11-s + (−0.286 − 0.957i)13-s + (0.686 − 0.727i)14-s + (0.597 − 0.802i)16-s + (−0.173 + 0.984i)17-s + (0.173 + 0.984i)19-s + (−0.893 − 0.448i)20-s + (−0.993 + 0.116i)22-s + (0.835 + 0.549i)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: $\chi_{81} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ 0.0774 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4656870131 + 0.4308955274i\)
\(L(\frac12)\) \(\approx\) \(0.4656870131 + 0.4308955274i\)
\(L(1)\) \(\approx\) \(0.5808460461 + 0.1041414149i\)
\(L(1)\) \(\approx\) \(0.5808460461 + 0.1041414149i\)

Euler product

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

−30.19295703671415166800605518309, −29.4126457107003429488251723219, −28.33660667866532159087900785138, −27.000773235509840306235729597885, −26.55871767296143453279133441283, −25.47463594415867221111317897383, −24.23193365891412757626175752847, −22.817444586295714735177250326264, −21.85121195275211532384852473447, −20.3011744463780849998343112275, −19.37861357783718919602692811875, −18.739076143765468359148293153142, −17.30101868650952083126554772907, −16.335653018941382039137201594747, −15.24081251031195343361343820197, −13.773513334231663183997066704399, −12.01788426885772688301747964994, −11.18793770681467739924534401039, −9.920770046737402818809600396338, −8.874514283956151019938281116690, −7.12354937666257815682999937456, −6.6997072472670195309968164198, −3.95137749682736156255856100295, −2.634989390469915839276864369750, −0.4604457722832379862233535522, 1.29471186090680582148909037649, 3.36953086564640590190026969635, 5.44561585315834121961524148105, 6.792766114450954548184238594807, 8.232778670934446672099159990394, 9.111783387269680233214474237503, 10.31117993015635407413054770479, 11.87267158796202334141974588799, 12.71210031901972427759385932847, 14.78892341726475106780941346470, 15.76888690148774708064686859638, 16.70179171026017669409837453189, 17.7144497944214960594953273609, 19.28267653100167861030130156704, 19.66739355025317910818936998580, 20.91960094895817131156199163575, 22.47320210354900780958223340089, 23.74071791867069206485832361565, 24.94599598428376042621102436616, 25.44910177709358242162416168161, 27.01735317518766425151033596199, 27.68604392379433547451916600069, 28.65275023707365828662525020366, 29.56830442460933496230653411681, 30.99039115132645279243600790272

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