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Properties

Label 1-3e4-81.29-r1-0-0
Degree $1$
Conductor $81$
Sign $-0.902 - 0.431i$
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  + (−0.396 − 0.918i)2-s + (−0.686 + 0.727i)4-s + (0.0581 − 0.998i)5-s + (0.973 − 0.230i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.835 − 0.549i)11-s + (−0.993 + 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.0581 − 0.998i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (0.686 + 0.727i)20-s + (−0.835 − 0.549i)22-s + (−0.973 − 0.230i)23-s + ⋯
L(s)  = 1  + (−0.396 − 0.918i)2-s + (−0.686 + 0.727i)4-s + (0.0581 − 0.998i)5-s + (0.973 − 0.230i)7-s + (0.939 + 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.835 − 0.549i)11-s + (−0.993 + 0.116i)13-s + (−0.597 − 0.802i)14-s + (−0.0581 − 0.998i)16-s + (−0.766 − 0.642i)17-s + (0.766 − 0.642i)19-s + (0.686 + 0.727i)20-s + (−0.835 − 0.549i)22-s + (−0.973 − 0.230i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(8.70465\)
Root analytic conductor: \(8.70465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (1:\ ),\ -0.902 - 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2583268342 - 1.139074551i\)
\(L(\frac12)\) \(\approx\) \(0.2583268342 - 1.139074551i\)
\(L(1)\) \(\approx\) \(0.6678347314 - 0.5984277815i\)
\(L(1)\) \(\approx\) \(0.6678347314 - 0.5984277815i\)

Euler product

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

−31.1646570196153020303114014701, −30.35191272526131673308543843094, −28.903841476347435580996115146362, −27.56183783733412394302248602703, −26.94969726771277654997696773193, −25.87281912159619353852436833578, −24.80247876789141721310502640802, −23.963298344820431674178445512218, −22.58538939526092324326025502151, −21.91700313018854562526236498809, −20.0664341089004748286766009090, −18.90925978029150914480081743005, −17.81431420446979037854976258963, −17.2022535858890728716430373421, −15.51220793515176249911365735333, −14.67743796180964349608914129888, −13.946596549748383623264794360056, −11.97960054333856667760170416942, −10.57668329100961377852504136577, −9.44725081953282417954039270636, −7.962947760062667425928496200673, −6.99955231323988985308989247907, −5.70098062155272639443765150168, −4.201466244152310100718022711265, −1.8782233733139503772346585024, 0.66173138861885638111543594550, 2.08795708363444206221167363527, 4.08056698040529720990890613237, 5.13505035698243131981134719679, 7.4999038227350354199921631305, 8.731676650925590413618986391670, 9.61924429293106264587182246176, 11.24849781696084308444501906500, 11.983729714988294103000233347375, 13.30848819104635817161238402329, 14.37289417352600042502724719111, 16.33428952952174162405765600225, 17.26722423414410820623298841744, 18.18311779898812632305236528178, 19.7798669535672498154000806153, 20.26275677837186692042753507974, 21.47604029330589890189548418942, 22.32709982776965126555932654224, 24.04981614804287410205434857948, 24.74875571595103643996124994837, 26.44346267348847708358369892787, 27.334983001160160098154087830576, 28.11499561135907319182025138006, 29.226104661467378163922874418609, 30.091618035518707735078971955620

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