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Properties

Label 1-189-189.130-r0-0-0
Degree $1$
Conductor $189$
Sign $-0.337 + 0.941i$
Analytic cond. $0.877712$
Root an. cond. $0.877712$
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  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.939 + 0.342i)20-s + (0.766 − 0.642i)22-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (0.173 − 0.984i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.939 + 0.342i)20-s + (0.766 − 0.642i)22-s + (0.766 − 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.337 + 0.941i$
Analytic conductor: \(0.877712\)
Root analytic conductor: \(0.877712\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ -0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9778795499 + 1.388860743i\)
\(L(\frac12)\) \(\approx\) \(0.9778795499 + 1.388860743i\)
\(L(1)\) \(\approx\) \(1.242404790 + 0.8836570766i\)
\(L(1)\) \(\approx\) \(1.242404790 + 0.8836570766i\)

Euler product

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

−27.40406018832863766940898322390, −25.424920567567810434100673514365, −24.92014435527692078423951736728, −23.808053843198760126762010145871, −22.95537740470565000593512575251, −22.11451402249225731711651583308, −20.81383819265117754675130690455, −20.45737718205507224978427742371, −19.51417799341831085083183355205, −18.235963376916538413160530845763, −17.181834630918201682223425545625, −15.876959569995209828110065705680, −15.03783196964532073132726525973, −13.83777365049080520437500553475, −12.84052577606663470276065840217, −12.29216510428448477442567069863, −11.05291485258063268501566498515, −9.89350341483754042663657522359, −9.00863359837715428281414124288, −7.44779189496834642022348592182, −5.94133507371473984620169951042, −4.9830535540454793093875598592, −4.01654045626068712265621739933, −2.48980978877575016307727382733, −1.15562889878905185003921341735, 2.34936775154321311659227649360, 3.49529880201091441396534120586, 4.65814181234681626274527062424, 6.26321898595358571269328662395, 6.615672816980227364739536795408, 8.03182884311637492670623517077, 9.12590508673348549177453160930, 10.8527991156861656526585011117, 11.503355185758465470303982208700, 12.94723995551032277599800187809, 13.84654446309521755137795039188, 14.643915122917117155987058105011, 15.54319874525861882336126837378, 16.64256554913019053191740065042, 17.54108410168715869166235557278, 18.69508566347167923502293090426, 19.6770555099990712526404770641, 21.46207782201514571047144935818, 21.54103143127584821544247659147, 22.737340592196962667866447126357, 23.59601763050005382203063494230, 24.46466954111224080550503915018, 25.48124823297552136804491708973, 26.49773515401945196264003469719, 26.7938768215222076516057480874

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