Properties

Degree 1
Conductor $ 3^{4} $
Sign $-0.902 - 0.431i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $-0.902 - 0.431i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (29, \cdot )$
Sato-Tate  :  $\mu(54)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (1:\ ),\ -0.902 - 0.431i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2583268342 - 1.139074551i$
$L(\frac12,\chi)$  $\approx$  $0.2583268342 - 1.139074551i$
$L(\chi,1)$  $\approx$  0.6678347314 - 0.5984277815i
$L(1,\chi)$  $\approx$  0.6678347314 - 0.5984277815i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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