Properties

Label 1-171-171.7-r0-0-0
Degree $1$
Conductor $171$
Sign $0.612 - 0.790i$
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  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)14-s + 16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.5 + 0.866i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.612 - 0.790i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.612 - 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.659927237 - 0.8131107847i\)
\(L(\frac12)\) \(\approx\) \(1.659927237 - 0.8131107847i\)
\(L(1)\) \(\approx\) \(1.614492454 - 0.4208975525i\)
\(L(1)\) \(\approx\) \(1.614492454 - 0.4208975525i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−27.90654132022001339089788554038, −26.42858319743014692980735817836, −25.58894038177382930637240232786, −24.81927633266485368007284276338, −23.45592853815448982054389374737, −22.85112170363517341367764803299, −22.13368841659627809293412298508, −21.03556698078512170244167795712, −20.08595684726776404226065044449, −18.92300164691436151633750925915, −18.1355663951735428865711166084, −16.43478479282785479615500939343, −15.36065335744412495068789280167, −15.08757452125146349272902176991, −13.65536425309214777225792768502, −12.75871912619137165531088060297, −11.63937090543781558896037983632, −10.89286493503505490723101936984, −9.51896361228397854770741406288, −7.83731367767455153471294941379, −6.79592388035669019622856807702, −5.834860232911976918463848556961, −4.47838322986526754486466137047, −3.19877118432611898981689192662, −2.28470707798163608695288542796, 1.260201102824247676572391281342, 3.2667756855718300213237575046, 4.091201127747600105935795325495, 5.318922607104936984764868717168, 6.47758601372074700811758436276, 7.7083434992293273458833805071, 8.85764808951152462648258366230, 10.63253230010284232738119871750, 11.28233665650591577903763882009, 12.85412510103349915295819663083, 13.115289254959769884133136668955, 14.31144202688143641593281195480, 15.69607717747209065707857676077, 16.249814054687903376815719548, 17.166028085730789752301890502923, 18.978267691180061107703260871419, 19.88893506075080792622265130014, 20.67815213159287913980662717957, 21.53585867169438308762296341147, 22.76961806956330293768523256428, 23.67388487134477351824106363308, 24.05474751877591039318663909064, 25.33080500309121750823819738463, 26.25706221819712203275186581364, 27.44754908872805191241820942315

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