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Properties

Label 1-171-171.4-r0-0-0
Degree $1$
Conductor $171$
Sign $0.672 + 0.740i$
Analytic cond. $0.794120$
Root an. cond. $0.794120$
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 + (−0.939 + 0.342i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.766 − 0.642i)23-s + (0.766 − 0.642i)25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.939 + 0.342i)5-s + 7-s + (−0.5 + 0.866i)8-s + (0.766 − 0.642i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.939 + 0.342i)14-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + (−0.5 + 0.866i)20-s + (0.766 + 0.642i)22-s + (0.766 − 0.642i)23-s + (0.766 − 0.642i)25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.672 + 0.740i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.672 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6331668679 + 0.2803628902i\)
\(L(\frac12)\) \(\approx\) \(0.6331668679 + 0.2803628902i\)
\(L(1)\) \(\approx\) \(0.6761075882 + 0.1599830948i\)
\(L(1)\) \(\approx\) \(0.6761075882 + 0.1599830948i\)

Euler product

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

−27.59773228874359816526152130258, −26.77561627630916439265425048153, −25.54700985195082065692200260363, −24.72904364119934838992859869444, −23.65556296557538034984023522917, −22.65780953097515984569363777779, −21.020897816642277057589498743879, −20.59188142061195743213967574033, −19.64335853012185047740046842143, −18.54956524026048695955996559757, −17.73570991039910364241359612449, −16.76922360781144319443533242886, −15.585686698796235638154242482596, −14.94684946449684351661674570671, −13.08479909034284741213857044671, −12.043272080042759983513745716229, −11.26400323056048485848929294869, −10.21381780770876783751357362857, −8.962819751188664922793546713035, −7.76621805233389662547700080552, −7.47192193346895414984369001498, −5.39445497217030236808354230846, −4.01233067882802289218159102634, −2.54882907363200954448686492762, −0.94522201789676153551429068571, 1.27798719603702888769069336630, 2.97114031567055852879169135583, 4.645899165049847202684574960107, 6.08578064999205083519480372960, 7.36356637393620064415912741058, 8.15651001939093460642310181983, 9.03675480341086876393044689020, 10.75413513014287444163332173855, 11.11280151882827459655208202900, 12.30320234443434368476787225517, 14.23720137578068177376891570502, 14.84291081923440843816379243317, 16.06456721537340568657782121721, 16.72144626546982072289426481029, 18.060763506282666787396382060385, 18.786132378561752041784655609395, 19.55736148086647780402409434425, 20.746040767140101757555452735366, 21.61451558174351388989209712405, 23.425474408998372774573520032125, 23.79169167133175323142164568019, 24.77038596758858813438658989985, 26.048296816813311023426767849860, 26.79266871005804332347443494338, 27.45176872202182149425119021208

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