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Properties

Label 1-171-171.16-r0-0-0
Degree $1$
Conductor $171$
Sign $0.188 - 0.982i$
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  + (0.766 − 0.642i)2-s + (0.173 − 0.984i)4-s + (0.766 − 0.642i)5-s + 7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)10-s + (−0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)14-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.5 − 0.866i)20-s + (0.173 + 0.984i)22-s + (0.173 − 0.984i)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 + 7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)10-s + (−0.5 + 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)14-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + (−0.5 − 0.866i)20-s + (0.173 + 0.984i)22-s + (0.173 − 0.984i)23-s + (0.173 − 0.984i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.472200473 - 1.216435073i\)
\(L(\frac12)\) \(\approx\) \(1.472200473 - 1.216435073i\)
\(L(1)\) \(\approx\) \(1.500681210 - 0.7922823756i\)
\(L(1)\) \(\approx\) \(1.500681210 - 0.7922823756i\)

Euler product

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

−27.31204833003230620802197920658, −26.71956120302593977799755080808, −25.61147659921150511080821986849, −24.767245676354059429046745897608, −24.01474666963255795984805939685, −22.93745262428877926058188362774, −21.89000220691364266484380380312, −21.34788581998815000351172994458, −20.35187153023610328237281503849, −18.66781440996737261150740263601, −17.732127404160690251521426949859, −16.97158327104181650666379290538, −15.70556489150432348216331878989, −14.65339261703294203434325022939, −14.01174992859948012014356779908, −13.09594979927897328215277718292, −11.709901691469754278315281825454, −10.81987547412759821768728882816, −9.31650602979668311963258297435, −7.90735740082866170335182269786, −7.09234043984372671332035011689, −5.65713293703447488713160801912, −5.04953921886665642576142926303, −3.3475781889855535376032335453, −2.21144956629988592320317634973, 1.554927701735068806749286874511, 2.42702326291407477151345182173, 4.385608332278435621357345655876, 5.02061680431562160756247854741, 6.19071365525048472641954343942, 7.77000633131316166152315280567, 9.28433536458382612535871938728, 10.19398781328399496334336006289, 11.29791445473605933851113038718, 12.51717052027169735132551486306, 13.08415467655109319685810695852, 14.49171213007842294190072866976, 14.89435486892500376601372573564, 16.53173188493076649487278226851, 17.6014232544762497018115381215, 18.57508204095415495657627722541, 19.94510427853357440652433796832, 20.682105537392881935225611710147, 21.425614022267980676973220738, 22.253475008502322703403414471812, 23.573525321060139349054725922399, 24.26650025693729714381579114913, 25.05208295202711793587170514892, 26.37684899882191953790535485504, 27.79874439033137979567648160509

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