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Properties

Label 1-171-171.106-r0-0-0
Degree $1$
Conductor $171$
Sign $0.884 - 0.466i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.884 - 0.466i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.884 - 0.466i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7940758964 - 0.1965463349i\)
\(L(\frac12)\) \(\approx\) \(0.7940758964 - 0.1965463349i\)
\(L(1)\) \(\approx\) \(0.8247709625 + 0.04178634678i\)
\(L(1)\) \(\approx\) \(0.8247709625 + 0.04178634678i\)

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

−27.97161115028109986427147664770, −26.58454414427503955566749948045, −25.782947493610688622509462541053, −25.160770711172915295147022275568, −23.692555305737783243704719540133, −22.23542049117036050103902333157, −21.771958035662910799645697374130, −20.89058544647307116467604307520, −19.76772997629231798755534678206, −18.82658108401116663418617321511, −17.90375795595758315355087427682, −17.18308489236734060693694619475, −15.95374793213344535459356856640, −14.57194846329136322060737937643, −13.27206222382788873727296818366, −12.584316355547443386871300712013, −11.50759208019432722978113047820, −10.121773114017500001852999854491, −9.54832741843866734930824260963, −8.55167555783675412621677347975, −7.05196273820200281534881683886, −5.662985195152691382104514477394, −4.27639722629058417679730363533, −2.58309455923726472989835628030, −1.87543394674714542032188569319, 0.799099629488144993301257873812, 2.74796530842060482906118190553, 4.66937375262808915568822359522, 5.80745176077873103133310246204, 6.73852261284606628573727087091, 7.8785311206235006517795624729, 9.09612157187156911214603250459, 10.11883550199837155387312717516, 10.77802092090153048184256355619, 12.877129954201354049937606167042, 13.71459365789445855044941928214, 14.476497331192118437691588347682, 15.93074573200182314845706242482, 16.59004289544250631477564934951, 17.64474391573910030392606457173, 18.302146628800390729684599278047, 19.56738813026676646919899960780, 20.47850789799266570458328401372, 21.89585914184838645336157093853, 22.76407597668567460331942353834, 23.80775229510852654928940981989, 24.80148771767372009641423247610, 25.47169893869266802509519244615, 26.61239123030564162168660465632, 26.97101664464088163985947143696

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