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Properties

Label 1-189-189.128-r1-0-0
Degree $1$
Conductor $189$
Sign $-0.178 + 0.983i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.766 + 0.642i)11-s + (0.766 + 0.642i)13-s + (0.173 + 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (−0.766 − 0.642i)5-s + (0.5 + 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.766 + 0.642i)11-s + (0.766 + 0.642i)13-s + (0.173 + 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)20-s + (−0.939 + 0.342i)22-s + (0.939 − 0.342i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $-0.178 + 0.983i$
Analytic conductor: \(20.3108\)
Root analytic conductor: \(20.3108\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 189,\ (1:\ ),\ -0.178 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.636835101 + 1.961031430i\)
\(L(\frac12)\) \(\approx\) \(1.636835101 + 1.961031430i\)
\(L(1)\) \(\approx\) \(1.504188475 + 0.6142324687i\)
\(L(1)\) \(\approx\) \(1.504188475 + 0.6142324687i\)

Euler product

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

−26.63590678851387582465252218426, −25.55868666332246732387069348254, −24.499682592373347168323868900361, −23.3646938989963430138979399757, −23.041561094704714747999795755697, −21.9023890909446332841878673488, −20.983652089453189941935920980190, −20.05528753791914078928801800834, −19.0117091485684679622355654815, −18.32518958282294432876261012843, −16.564165762599739122339284341118, −15.518674736376673529339130072075, −14.99990866229268465762043233441, −13.661309687059580847136808845719, −12.98386508754900835997514059570, −11.52799394084346865279531101852, −11.09433012148266078273389388708, −9.94906562601026941762971778633, −8.19104156863911100125015499128, −7.10379314733227898878256574234, −5.94477668166739753861090347170, −4.76135607218117234403099753665, −3.43452031629874034334182738969, −2.67692767755441853584029861649, −0.657242239986933797136323663581, 1.66837269211979493930548355155, 3.37786352130767886550790004580, 4.366592701047112161352538659614, 5.3692656105283723177927861848, 6.66962630733445885274585870063, 7.82959346882685991637616312020, 8.662555898326897000674025467649, 10.45998712777316558377347971801, 11.57198539272192959823279199438, 12.56637368355343220600568837539, 13.18634836171557334534619292229, 14.57889001679171040863799546534, 15.36086967927403028671568205356, 16.33367737918032200387851866610, 17.02106502864627085990878713950, 18.56464947987727127602837554158, 19.70740504579599111958991485322, 20.80248993806895502706626234134, 21.23164401815734733344003747969, 22.71602450757401448616740970117, 23.44957088927523098010762193298, 23.97653100701297774900512580140, 25.17525440560230528201196550767, 25.93181905615704774644675291022, 27.056898862223381561349793993378

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