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Properties

Label 1-163-163.16-r0-0-0
Degree $1$
Conductor $163$
Sign $0.557 + 0.829i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (0.987 + 0.154i)2-s + (−0.999 + 0.0387i)3-s + (0.952 + 0.305i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.323 − 0.946i)7-s + (0.893 + 0.448i)8-s + (0.996 − 0.0774i)9-s + (−0.790 + 0.612i)10-s + (0.533 + 0.845i)11-s + (−0.963 − 0.268i)12-s + (−0.0581 + 0.998i)13-s + (0.466 − 0.884i)14-s + (0.657 − 0.753i)15-s + (0.813 + 0.581i)16-s + (−0.835 + 0.549i)17-s + ⋯
L(s)  = 1  + (0.987 + 0.154i)2-s + (−0.999 + 0.0387i)3-s + (0.952 + 0.305i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.323 − 0.946i)7-s + (0.893 + 0.448i)8-s + (0.996 − 0.0774i)9-s + (−0.790 + 0.612i)10-s + (0.533 + 0.845i)11-s + (−0.963 − 0.268i)12-s + (−0.0581 + 0.998i)13-s + (0.466 − 0.884i)14-s + (0.657 − 0.753i)15-s + (0.813 + 0.581i)16-s + (−0.835 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.557 + 0.829i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.557 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.274321444 + 0.6788779676i\)
\(L(\frac12)\) \(\approx\) \(1.274321444 + 0.6788779676i\)
\(L(1)\) \(\approx\) \(1.299154343 + 0.3725099423i\)
\(L(1)\) \(\approx\) \(1.299154343 + 0.3725099423i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.987 + 0.154i)T \)
3 \( 1 + (-0.999 + 0.0387i)T \)
5 \( 1 + (-0.686 + 0.727i)T \)
7 \( 1 + (0.323 - 0.946i)T \)
11 \( 1 + (0.533 + 0.845i)T \)
13 \( 1 + (-0.0581 + 0.998i)T \)
17 \( 1 + (-0.835 + 0.549i)T \)
19 \( 1 + (0.856 + 0.516i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.627 - 0.778i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
37 \( 1 + (0.396 - 0.918i)T \)
41 \( 1 + (0.952 - 0.305i)T \)
43 \( 1 + (0.925 - 0.378i)T \)
47 \( 1 + (-0.360 - 0.932i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.893 - 0.448i)T \)
67 \( 1 + (-0.740 - 0.672i)T \)
71 \( 1 + (-0.135 + 0.990i)T \)
73 \( 1 + (-0.740 + 0.672i)T \)
79 \( 1 + (-0.565 + 0.824i)T \)
83 \( 1 + (-0.981 + 0.192i)T \)
89 \( 1 + (0.533 - 0.845i)T \)
97 \( 1 + (-0.431 - 0.902i)T \)
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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

−27.860470255736932791086326806967, −26.991551778680037477622812740813, −24.962610274472407431520094995327, −24.45904530801088787867322911101, −23.764584419692900873548748768028, −22.47504899260587975924947491768, −22.1372704561530842475410613424, −20.93569904345435442453546568230, −19.97201689323488149397222781149, −18.825601226292075432583329758790, −17.61178632907934959905159742283, −16.22301119065797853099768232661, −15.83532222896976252966201716073, −14.6755562407783346063133963555, −13.147961161028358579050991715951, −12.39846511214604189589331718644, −11.53075971523301414676745333927, −10.892046710925464117457464855210, −9.118934240126798393502555253400, −7.6594390425578212145198926474, −6.30749426932811641949448369380, −5.28686871101333295686268008323, −4.56966603619368974750008066644, −3.03513691659587047609953822132, −1.14089407087697564789360974513, 1.81329008450055442311161561063, 3.93104772260117023042155009103, 4.330621867517097639348893158471, 5.89214132652760597047738674499, 7.053273043550237409390665093018, 7.491603063030492695743965553993, 9.89099652734367321451585637249, 11.18454048437495386842089448589, 11.508723033245893328592284482305, 12.69557862049844884626939092818, 13.92331791439112474516078651807, 14.91055187238281751001543230708, 15.86572296605922199249938935364, 16.87263408894037734952308543083, 17.728457375092402675407253125510, 19.19673830876402495424932560045, 20.2200072359823015704372078914, 21.37194793898600422780676059051, 22.41511242934454215858776202042, 22.95131483325106957998630464986, 23.77701957765746962729310177232, 24.475375996754580870846885271735, 26.057817577068427962201804997956, 26.83375546679032611199858104264, 27.99055355331876066223119031116

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