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Properties

Label 1-189-189.149-r1-0-0
Degree $1$
Conductor $189$
Sign $0.838 - 0.545i$
Analytic cond. $20.3108$
Root an. cond. $20.3108$
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 + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)8-s + 10-s + (−0.766 − 0.642i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s − 17-s + 19-s + (−0.766 − 0.642i)20-s + (0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.173 + 0.984i)4-s + (−0.766 + 0.642i)5-s + (0.5 − 0.866i)8-s + 10-s + (−0.766 − 0.642i)11-s + (−0.939 − 0.342i)13-s + (−0.939 + 0.342i)16-s − 17-s + 19-s + (−0.766 − 0.642i)20-s + (0.173 + 0.984i)22-s + (0.939 + 0.342i)23-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.838 - 0.545i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ 0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7627272847 - 0.2261874150i\)
\(L(\frac12)\) \(\approx\) \(0.7627272847 - 0.2261874150i\)
\(L(1)\) \(\approx\) \(0.6113704101 - 0.1118955111i\)
\(L(1)\) \(\approx\) \(0.6113704101 - 0.1118955111i\)

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 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 - T \)
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

−26.78328866825667883627970433659, −26.32927829857628833355356749456, −24.89200849640460984239905960008, −24.36965982966917354439224039637, −23.42019495018948087500750105922, −22.55690221392168736361948426113, −20.92283817199296508049673966380, −19.99063888850391838918986555099, −19.30704544857549477137875739010, −18.1513423645034334140977112064, −17.26570305245023604138195862361, −16.235173778400178676682962253232, −15.519739473040575730888011506860, −14.61198217302857330712953157109, −13.21727287003691432304946077524, −12.01685065477174900100807493685, −10.91008833103099857386304023584, −9.693759797037082730549158191745, −8.78093816501198216961875512153, −7.674681254697537775347392897907, −6.95491149468965218576007927357, −5.30667937419150790698468863306, −4.476544378252546595983165149240, −2.38575382645200770278725422606, −0.6953353468561053651632452058, 0.60951242745531152411411391646, 2.54773078084338911189623476196, 3.360421421592681368343546854854, 4.86370339791525868749482058318, 6.7830621921724022951903535461, 7.681214334562814220604462803913, 8.62506309883005529302732115959, 9.9461511611626171939703866113, 10.86631176820868163940623946866, 11.6598787111574961028687759541, 12.70910668418988934468735659661, 13.91128064016747553895539158868, 15.4027248695561236051805817695, 16.07541164516212634069787737190, 17.41179301605054459002277285371, 18.222496182324037071190099886565, 19.22105304021727311773902497157, 19.79697995129386732016199728364, 20.92784125475798897463656404362, 21.97347655420985613382126847571, 22.73142604544728596416392102267, 24.024511081477747786588002549787, 25.06802787028668207638015812707, 26.340049375324716675473650709400, 26.84544748716017485532162196059

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