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Properties

Label 1-189-189.67-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  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)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 + (0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.5 + 0.866i)8-s + (−0.5 + 0.866i)10-s + (−0.939 + 0.342i)11-s + (−0.939 − 0.342i)13-s + (0.766 + 0.642i)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 + (0.173 + 0.984i)23-s + (0.766 + 0.642i)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: $\chi_{189} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (0:\ ),\ -0.337 + 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03646586863 - 0.05179166843i\)
\(L(\frac12)\) \(\approx\) \(-0.03646586863 - 0.05179166843i\)
\(L(1)\) \(\approx\) \(0.4996628610 - 0.3079485612i\)
\(L(1)\) \(\approx\) \(0.4996628610 - 0.3079485612i\)

Euler product

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

−27.28109849994625486444457729509, −26.821582401446054660924251580705, −25.98872634216868925313854726528, −24.76698985715744356735458530617, −24.02197897114436812836292517243, −23.14639829396091471265571525485, −22.412496584138837689491425092382, −21.346931750816300415046975633851, −20.04086051604634790290901140322, −18.764873543647350231523020741705, −18.27274776821414257500363156567, −16.78334885597387582764042439317, −16.148730299858771463205587732752, −15.08403712998907415259816783750, −14.42859552259583480187732328121, −13.146899120046982021633013684847, −12.188631404874149883247806590407, −10.9203294711518524260884715180, −9.583567155564749431091028069996, −8.29583421090838525865707010038, −7.52812030909594964778668017200, −6.54664470432705086992121854816, −5.14806784159409067425212197391, −4.160889748062409519232193341315, −2.81455478142214576436660596996, 0.04483286576142395788838680880, 2.00016597997446386030342167156, 3.32800580423254828383789960899, 4.49103701385534866049139355053, 5.40363814540919043312635996406, 7.34246329188642074340470971671, 8.400072716346906804188341865597, 9.52384796196268119170628227433, 10.7120377050318004788274846927, 11.51622332214065860057578030441, 12.70261779178035892989479198898, 13.15068035101916644826915908702, 14.81010282110731130770364249191, 15.4279529220058328630440897568, 16.949349822752846838793462109427, 17.94479681265908026405464763644, 19.08834689329370997146956750193, 19.804184725139064199784670819860, 20.54842753524697363276634140794, 21.67563361008746679022853408815, 22.49974645744519896866615488217, 23.73365277975785818567517132818, 23.97380840247694342945442407626, 25.7474585301103439128353027722, 26.78131604346138097195701699227

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