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Properties

Label 1-163-163.33-r0-0-0
Degree $1$
Conductor $163$
Sign $0.254 - 0.966i$
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  + (0.856 − 0.516i)2-s + (−0.135 + 0.990i)3-s + (0.466 − 0.884i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (−0.360 − 0.932i)7-s + (−0.0581 − 0.998i)8-s + (−0.963 − 0.268i)9-s + (−0.740 − 0.672i)10-s + (0.925 − 0.378i)11-s + (0.813 + 0.581i)12-s + (−0.835 − 0.549i)13-s + (−0.790 − 0.612i)14-s + (0.987 − 0.154i)15-s + (−0.565 − 0.824i)16-s + (0.893 + 0.448i)17-s + ⋯
L(s)  = 1  + (0.856 − 0.516i)2-s + (−0.135 + 0.990i)3-s + (0.466 − 0.884i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (−0.360 − 0.932i)7-s + (−0.0581 − 0.998i)8-s + (−0.963 − 0.268i)9-s + (−0.740 − 0.672i)10-s + (0.925 − 0.378i)11-s + (0.813 + 0.581i)12-s + (−0.835 − 0.549i)13-s + (−0.790 − 0.612i)14-s + (0.987 − 0.154i)15-s + (−0.565 − 0.824i)16-s + (0.893 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.254 - 0.966i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.254 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220465734 - 0.9403695247i\)
\(L(\frac12)\) \(\approx\) \(1.220465734 - 0.9403695247i\)
\(L(1)\) \(\approx\) \(1.338097025 - 0.5298588846i\)
\(L(1)\) \(\approx\) \(1.338097025 - 0.5298588846i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.856 - 0.516i)T \)
3 \( 1 + (-0.135 + 0.990i)T \)
5 \( 1 + (-0.286 - 0.957i)T \)
7 \( 1 + (-0.360 - 0.932i)T \)
11 \( 1 + (0.925 - 0.378i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.893 + 0.448i)T \)
19 \( 1 + (0.323 + 0.946i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.0193 - 0.999i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
37 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + (0.466 + 0.884i)T \)
43 \( 1 + (-0.211 - 0.977i)T \)
47 \( 1 + (-0.875 - 0.483i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (0.533 - 0.845i)T \)
71 \( 1 + (0.952 + 0.305i)T \)
73 \( 1 + (0.533 + 0.845i)T \)
79 \( 1 + (0.249 - 0.968i)T \)
83 \( 1 + (-0.627 + 0.778i)T \)
89 \( 1 + (0.925 + 0.378i)T \)
97 \( 1 + (0.713 + 0.700i)T \)
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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

−28.08081098688703956731290545394, −26.60651337383206952025921596016, −25.66923060035954568833983372635, −24.90000140330341493388995168586, −24.13424141475904663771197988116, −22.94685643202826788124900628107, −22.43649907504330401829867914094, −21.58531334692683468208787368032, −19.9451980555306010945375187863, −19.065279277532469383037729888651, −18.06641637705792524930543680110, −17.039654240630407273619592843572, −15.85310283862259848003863270709, −14.5632112953971236488807680329, −14.25095896445634165254043175113, −12.725382045008487280932331586741, −12.02349354260389668662159522688, −11.25447807061752722465052387619, −9.27977354783093828263153468670, −7.84470255638544335334562705057, −6.862647398493311573618989111446, −6.28034580713558963572379452876, −4.8951808950635083575691522716, −3.15442337178994578352710267586, −2.2862297558624479531212811877, 1.08267837082137825625839199125, 3.311787925093865612033245041432, 4.063140883185145045244153899051, 5.086140705425055263667875330478, 6.15808294003059655701923975296, 7.91617578467462875272321530822, 9.5869701597305238945827679853, 10.11888776659241781039623212056, 11.488225544472459476178485224027, 12.24305825646579332524345838184, 13.49101788959971086172156753911, 14.476495613078443562656315306206, 15.49475927101433218880954080668, 16.57337737937425453450298192860, 17.135321398759083147195635195339, 19.36366477306875602616741031246, 19.91869345098788024412861058291, 20.79784276168119854499905396889, 21.59117574903049442870210602409, 22.715727524473738451890647759842, 23.293302852687196971820650751994, 24.452516695995371662507364292917, 25.4157963519884481084650524519, 27.04340141341515813884487545264, 27.51071777257356609576145471424

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