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Properties

Label 1-171-171.22-r1-0-0
Degree $1$
Conductor $171$
Sign $-0.174 + 0.984i$
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  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.173 + 0.984i)10-s + 11-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + (−0.5 + 0.866i)20-s + (0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.173 + 0.984i)5-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.173 + 0.984i)10-s + 11-s + (−0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + (−0.5 + 0.866i)20-s + (0.939 + 0.342i)22-s + (0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ -0.174 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.089820855 + 2.492202823i\)
\(L(\frac12)\) \(\approx\) \(2.089820855 + 2.492202823i\)
\(L(1)\) \(\approx\) \(1.706606227 + 0.8904157187i\)
\(L(1)\) \(\approx\) \(1.706606227 + 0.8904157187i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 - T \)
37 \( 1 - T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−27.51568537964912610812201739245, −25.55858151642842001949120304856, −24.85935987226312159008618571188, −24.32051104412661800034424123613, −22.8823264130231152427682610351, −22.29505527607410420949167497565, −21.25252866479518169075261923218, −20.32552959379501167350006045524, −19.5947442579895469804687233610, −18.40801697962881175911822399669, −16.912688729202327781091888187261, −16.00703163993188847648707729422, −15.05103966365154324650731770088, −13.962681896227163628478687120175, −12.72128706962170383076131771402, −12.35349849549124087374888213458, −11.15681480323284567300291247775, −9.701648646103918978946052590657, −8.85282559125235477036433980910, −7.12891855084157640296816291378, −5.78785165139949972492871547304, −5.0778524496895427287115976667, −3.688256526856289555648562128733, −2.39508092395337853815933213784, −0.88341193734925810494926960525, 1.87198673261398216788372045424, 3.44584254080619264857686990316, 4.13428093991733017479865822660, 5.87695426944775073250368731144, 6.79116095835113136509252918189, 7.48523655789046824953505099927, 9.287737103921721042480961122500, 10.658713253305575593557807667018, 11.49620655765271278783371775563, 12.74768486027787295578663046257, 13.87172701315208664116767450100, 14.4545645584384650744860589102, 15.4674888202947997671951931379, 16.77994413604383695193630082408, 17.31994549677056697261645948108, 19.03525727973926911093172633410, 19.77404535685078991433787147529, 21.10201626647858583594124193435, 21.97200763945679688097920138360, 22.74683156960866022639721412517, 23.541560906803160781000420220734, 24.53549843331838447386565278494, 25.80041584376626859666221554885, 26.1891730244633760955689528863, 27.30839151336697233790517810187

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