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Properties

Label 1-171-171.85-r0-0-0
Degree $1$
Conductor $171$
Sign $-0.755 - 0.654i$
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  + (0.173 − 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

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

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ -0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4037884833 - 1.082699680i\)
\(L(\frac12)\) \(\approx\) \(0.4037884833 - 1.082699680i\)
\(L(1)\) \(\approx\) \(0.7918510177 - 0.7684180406i\)
\(L(1)\) \(\approx\) \(0.7918510177 - 0.7684180406i\)

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

−27.5989631970038167547362726856, −26.7214435886391713972901026285, −25.91864389694193134170901442981, −25.166026020737379650120941446012, −23.90687289139174485501503040685, −23.35079179342488574099238460048, −22.20847615177759662371654518629, −21.47602319469116639138380681393, −20.24169326505539117556605395733, −18.51856999651853769886046001847, −18.123898629597257467925658353847, −17.216105576561675580901276378200, −15.82544094679816021765645941069, −15.06332013087218867750287126231, −14.15333274590340288592202969058, −13.38319289681115188200522651599, −11.86273354089909594022713518608, −10.70883019775710145896367449632, −9.45542216664442248193477267280, −8.18661649808792721223314478664, −7.22972122631688879069057055687, −6.278371562488020113268415497674, −4.98593627001330552088203795740, −3.87830743304845430648209483227, −2.12625987351938180237892485408, 0.97245873523740972850116992026, 2.25575863548343157100260468120, 3.87635261358538909145712859195, 4.95227335270954863670015478704, 5.890546360098652856760601061128, 8.3066449252163165502606296934, 8.57100551820341860647885118631, 10.12756647554331718881580972842, 11.08834838380169542366391619525, 12.019210163813364503488468019237, 13.20159370815000491846436552004, 13.77093351387040536508188242456, 15.135066264054358128359452334263, 16.41387740590570804418722424634, 17.71425159710997850819757395881, 18.26859603226538018376687533206, 19.69976177994045821918039052504, 20.43884307839780925229359537820, 21.24806331030790166231072664854, 21.9508145192344231409106874906, 23.44392148042454167961565854664, 24.01672653514268218296748078643, 25.05674521015032642347028238820, 26.59790793410024817934191449548, 27.37542030907784623056824337168

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