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Properties

Label 1-171-171.74-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.654 - 0.755i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
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 + (−0.173 − 0.984i)5-s + 7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (0.5 + 0.866i)20-s + (−0.939 − 0.342i)22-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.173 − 0.984i)5-s + 7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)10-s + (0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (0.5 + 0.866i)20-s + (−0.939 − 0.342i)22-s + (0.939 − 0.342i)23-s + (−0.939 + 0.342i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.654 - 0.755i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.654 - 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7130127930 - 1.561154215i\)
\(L(\frac12)\) \(\approx\) \(0.7130127930 - 1.561154215i\)
\(L(1)\) \(\approx\) \(0.8411690891 - 0.6950717489i\)
\(L(1)\) \(\approx\) \(0.8411690891 - 0.6950717489i\)

Euler product

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

−27.48445095477153477426960653620, −26.69231885813824831379059940396, −25.50431234193692228241808959359, −25.08728707538667041286443064943, −23.66549135098664122299846617966, −23.07401328039102515166614580746, −22.17443896082576508784039620054, −20.96504257636450861614347957923, −19.635977364765927731855517480386, −18.39295277822113284948349342112, −17.94668784952902495732151348682, −16.89665406929902972597154741433, −15.61321701971615547808577686544, −14.733232543812085275034229425286, −14.24238448729303812412280386669, −12.85696944227903240194100758090, −11.36921340594865137175543198638, −10.35401543354169155869194590920, −9.14208650881197670773876767597, −7.83542473627632431694871570706, −7.18455637849640272259487582965, −5.92680114452695757858130270729, −4.75398813024938790601179864409, −3.39649275127566493072957138801, −1.353795816012385080165421662630, 0.80216041313632099722397602403, 1.761006193692774695202548083083, 3.58973162143411414392973463725, 4.57053836099490234459020244684, 5.741613633000491571860350655154, 7.8810321230715976319416235658, 8.637191207811242883960158645302, 9.54910651672876881434772124489, 11.123041787219670800103840000504, 11.59538571570878665450480968935, 12.79864723610086540051302983580, 13.73055871196610533092511639568, 14.78203174896643905980945052966, 16.54137594635235160213258646470, 17.05934179817577009931912269532, 18.38813092264464397183403167747, 19.17388551959051393978682495374, 20.309145370423381305752742040733, 21.06313637555706430921198623833, 21.64550094246218307437103446784, 23.12132018136250500600024716204, 23.935187789893255308618993803687, 24.92005028076905495484327470865, 26.30252489059112114134672969785, 27.32287122357123330454753477933

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