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Properties

Label 1-3e3-27.2-r1-0-0
Degree $1$
Conductor $27$
Sign $0.686 + 0.727i$
Analytic cond. $2.90155$
Root an. cond. $2.90155$
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  + (0.939 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.686 + 0.727i$
Analytic conductor: \(2.90155\)
Root analytic conductor: \(2.90155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 27,\ (1:\ ),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.071104870 + 0.8933874351i\)
\(L(\frac12)\) \(\approx\) \(2.071104870 + 0.8933874351i\)
\(L(1)\) \(\approx\) \(1.693468257 + 0.5069911147i\)
\(L(1)\) \(\approx\) \(1.693468257 + 0.5069911147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−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