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Properties

Label 1-171-171.50-r0-0-0
Degree $1$
Conductor $171$
Sign $0.0765 - 0.997i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s − 22-s + (0.5 − 0.866i)23-s + 25-s − 26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (−0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)20-s − 22-s + (0.5 − 0.866i)23-s + 25-s − 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0765 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0765 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.0765 - 0.997i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.0765 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4825515365 - 0.4469165884i\)
\(L(\frac12)\) \(\approx\) \(0.4825515365 - 0.4469165884i\)
\(L(1)\) \(\approx\) \(0.6263403248 - 0.2872065917i\)
\(L(1)\) \(\approx\) \(0.6263403248 - 0.2872065917i\)

Euler product

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

−27.72531489403454999512058497934, −26.58179538850658588446222806358, −26.05218966064226490413152055557, −24.98638461227654207388578294685, −23.69594302068832637115748219865, −23.33791844484736632636037446113, −22.43303619214139999317032619704, −20.75017708611039991723040803801, −19.42947150083605505788020376890, −19.28050151698716213660694312394, −17.75771990942427169415960300932, −16.83027049915270765222162450574, −16.035469561830950873908881582287, −15.07700622634841497084069739728, −14.12601436120423277052703433061, −12.92512575693523093016916134767, −11.54503283804357862627941429202, −10.35858903533026066300883175554, −9.33049280276970049391677562794, −8.117710094962585102268165497158, −7.17557254905911146911683551855, −6.36259665038787934652620793069, −4.62095360528706193201016931737, −3.74598276571722351294532936176, −1.267760763474928088053098986649, 0.781811041936933921815328707795, 2.833362448537129447513168686200, 3.55551234844273731939568596748, 5.095783076868360984810548467764, 6.78195675871218027167793736354, 8.257659330925692324841755028461, 8.795476376773137452536274551478, 10.157622622828129326881807542046, 11.28031193183009531824452155218, 12.067794964239477274835639436874, 12.91275164638125628431698041551, 14.27574342518004349099290099069, 15.76498937987299867896890195892, 16.37757672830650196933790432753, 17.80174911736504111876367205848, 18.83592481537612949596264007092, 19.33184284781800300769811935403, 20.37574312802457982333084753120, 21.35999296616309630467430451471, 22.49224410245206215506509174470, 23.03087713654108669187469879672, 24.62146793975310302761761730518, 25.49488668017378120874512063550, 26.73942600114682909620056554956, 27.40075556837698865550840733435

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