Properties

Degree 1
Conductor $ 3^{4} $
Sign $0.996 + 0.0774i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.993 + 0.116i)2-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.0581 + 0.998i)11-s + (0.597 − 0.802i)13-s + (0.396 + 0.918i)14-s + (0.893 − 0.448i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.973 + 0.230i)20-s + (−0.0581 − 0.998i)22-s + (−0.286 + 0.957i)23-s + ⋯
L(s,χ)  = 1  + (−0.993 + 0.116i)2-s + (0.973 − 0.230i)4-s + (0.893 + 0.448i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (−0.939 − 0.342i)10-s + (−0.0581 + 0.998i)11-s + (0.597 − 0.802i)13-s + (0.396 + 0.918i)14-s + (0.893 − 0.448i)16-s + (0.766 − 0.642i)17-s + (0.766 + 0.642i)19-s + (0.973 + 0.230i)20-s + (−0.0581 − 0.998i)22-s + (−0.286 + 0.957i)23-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(81\)    =    \(3^{4}\)
\( \varepsilon \)  =  $0.996 + 0.0774i$
motivic weight  =  \(0\)
character  :  $\chi_{81} (40, \cdot )$
Sato-Tate  :  $\mu(27)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 81,\ (0:\ ),\ 0.996 + 0.0774i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7338024755 + 0.02847487788i$
$L(\frac12,\chi)$  $\approx$  $0.7338024755 + 0.02847487788i$
$L(\chi,1)$  $\approx$  0.7903779748 + 0.02932987410i
$L(1,\chi)$  $\approx$  0.7903779748 + 0.02932987410i

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

−30.78110116871050085088573258099, −29.55843493575035940891115885841, −28.63448449628008300666598789848, −28.11961842634769876439692588400, −26.64269929948755088778907694473, −25.7179928175184091725595183078, −24.85171023255120972072118587464, −23.91078449138515202423166479870, −21.77488724742358337747076942767, −21.34316873940910365808346058164, −19.99190041316430376601914446415, −18.73791081306528977834774111779, −18.06608552547573329341162282728, −16.641267081011651344186289704545, −16.032036522730955179680261681523, −14.38628231788394195260542529329, −12.87894472652382113220280717974, −11.69868418623004272543066613664, −10.3679088021711185390025732372, −9.11425568973302878511564615568, −8.48773235132545478514735362599, −6.56941601463254818571318780825, −5.559741354438408090208429253854, −3.05343002450603353015584344131, −1.54887701770977937442511451377, 1.50120548673492442729048298941, 3.21123333482134676854408223536, 5.586847104583670306328294024113, 6.89628939201845493675705672781, 7.89194634722407838512945088082, 9.8182859348706313353223055664, 10.05368943902825942482520394057, 11.53681914402553463982594267839, 13.206340126145945756240228954507, 14.4673870399616747121479667503, 15.773593194345885757929914689957, 17.01482130041922819895879682815, 17.818166412991566444553231833584, 18.76473126175363901146504437265, 20.2417098890790849125606735606, 20.8037495500588811347322757498, 22.484859165468284118324012821695, 23.56116326309607292273817155893, 25.19255019685459071374540164582, 25.614554972445407143602838352839, 26.71221259249183400693154364271, 27.72925707370024731637561482914, 28.93848123062537913171495593609, 29.72646037262166390415007887830, 30.55701779624211275085959815417

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