Properties

Label 1-171-171.151-r1-0-0
Degree $1$
Conductor $171$
Sign $0.766 - 0.642i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (−0.5 − 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + (−0.5 − 0.866i)20-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ 0.766 - 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8891851025 - 0.3236369100i\)
\(L(\frac12)\) \(\approx\) \(0.8891851025 - 0.3236369100i\)
\(L(1)\) \(\approx\) \(0.9030205325 + 0.3286725947i\)
\(L(1)\) \(\approx\) \(0.9030205325 + 0.3286725947i\)

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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + T \)
23 \( 1 + (-0.5 + 0.866i)T \)
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 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
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

−28.02186028723251582399137892728, −26.61833463556177400859935764538, −25.336085684415408613952660779263, −24.30101594343430839750008213519, −23.32748597053188037026557947100, −22.65975369079663304650750374351, −21.28733047418820331671529263919, −20.832633469959521693231529587490, −19.64273322734451100853293020057, −18.92619380042927595259454669952, −17.89982728685744377189747764103, −16.31995156898386475833860747654, −15.498802257472227786697319795818, −14.35445662386233576589561027449, −13.0770492113864910745008970335, −12.29900291908717180768812373643, −11.67889602494808519306103436174, −10.16382230374122978200885689854, −9.24632814631317405998189008637, −8.19298024538805857764961641996, −6.35036750544842780389774707038, −5.13036796311919170328579476012, −4.184184350055567962583556254733, −2.80127303762705211794058000888, −1.4161510475773265526464407584, 0.302353214030493977628937531333, 3.18162076485516617842880190244, 3.70142986252540151326684144136, 5.39798074432414684999301822360, 6.462065742097177797197550152105, 7.51973345440327828903868580705, 8.27021680487133371488547745471, 9.984717256631673875531262668373, 11.034548184436482169926642132081, 12.356505354543266100683686503174, 13.54718586354016344418567204501, 14.18371041067894424942122236290, 15.451571442679535033538674979098, 16.07592517254436788173753629980, 17.17709014934068286873983456510, 18.28177669836494704561303702878, 19.21293107156440324427394849964, 20.53000025648083990946172480772, 21.68749104877824176686112308875, 22.6691915610629698957867511982, 23.33000574782244932067448316434, 24.04302373712285554637563280184, 25.47812291272055701105364605520, 26.07414210454181284143896918681, 26.93228696153651096999666799515

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