Properties

Degree 1
Conductor $ 3^{3} $
Sign $0.686 + 0.727i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.173 − 0.984i)11-s + (−0.939 + 0.342i)13-s + (0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.766 − 0.642i)23-s + ⋯
L(s,χ)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (0.766 − 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.173 − 0.984i)11-s + (−0.939 + 0.342i)13-s + (0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.766 + 0.642i)20-s + (0.173 − 0.984i)22-s + (−0.766 − 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.686 + 0.727i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.686 + 0.727i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(27\)    =    \(3^{3}\)
\( \varepsilon \)  =  $0.686 + 0.727i$
motivic weight  =  \(0\)
character  :  $\chi_{27} (2, \cdot )$
Sato-Tate  :  $\mu(18)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 27,\ (1:\ ),\ 0.686 + 0.727i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.071104870 + 0.8933874351i$
$L(\frac12,\chi)$  $\approx$  $2.071104870 + 0.8933874351i$
$L(\chi,1)$  $\approx$  1.693468257 + 0.5069911147i
$L(1,\chi)$  $\approx$  1.693468257 + 0.5069911147i

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

−37.5133816611696645329604491317, −36.31841020614197660949147835154, −34.564308501135083620274381586638, −33.39356908290097460807293025414, −32.03322842928821457711962847547, −31.26567410102111338560968710387, −29.95543595323483945612623839767, −28.46321836907213471252069700940, −27.67340199926699433354801911443, −25.271109151244888044954697463849, −24.320587100993145732504595963955, −23.16814170117636041896505813731, −21.58064217699438680979435816553, −20.62589196493439134091083736182, −19.378554976055638172940211534073, −17.3680276979290959787984954896, −15.63388976653377013031231855848, −14.50437732966114659069477724520, −12.68007819388464012863046390259, −11.9355147367484747346440851740, −9.991470414909572147048294478793, −7.91438509969654085095805260558, −5.601706788614045471861445311395, −4.36520335611636501165930356609, −1.92612712073415051907750420112, 2.860516751384937779069980089754, 4.67511275894351686242751712383, 6.60501479198176008537476565692, 7.89316727903529802303258863456, 10.664298392187542185048700515350, 11.83148258646725015257941183458, 13.82344868704737974178134617838, 14.58147314594082583881212134431, 16.13394544126673928874690926553, 17.65233965292251957060418623829, 19.474105673837352034479987755721, 21.10207807132164329616702587115, 22.1926824742599547555411765847, 23.48146292953663819578070443289, 24.490886264181400886607138246431, 26.14393830784178631658583858488, 27.08801668705320388410210023406, 29.4084404225114357787276165107, 30.20374539284429544169813633525, 31.36834947262606727605218485366, 32.64280915195817769583370829302, 34.15019652323261997852977506671, 34.42007554618066471941828844409, 36.44802792303360602147236236885, 37.95795327445094493223772281054

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