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Properties

Label 1-163-163.83-r0-0-0
Degree $1$
Conductor $163$
Sign $0.000518 + 0.999i$
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  + (−0.740 − 0.672i)2-s + (−0.565 + 0.824i)3-s + (0.0968 + 0.995i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)6-s + (0.925 + 0.378i)7-s + (0.597 − 0.802i)8-s + (−0.360 − 0.932i)9-s + (0.713 − 0.700i)10-s + (0.996 + 0.0774i)11-s + (−0.875 − 0.483i)12-s + (−0.993 + 0.116i)13-s + (−0.431 − 0.902i)14-s + (−0.790 − 0.612i)15-s + (−0.981 + 0.192i)16-s + (0.396 + 0.918i)17-s + ⋯
L(s)  = 1  + (−0.740 − 0.672i)2-s + (−0.565 + 0.824i)3-s + (0.0968 + 0.995i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)6-s + (0.925 + 0.378i)7-s + (0.597 − 0.802i)8-s + (−0.360 − 0.932i)9-s + (0.713 − 0.700i)10-s + (0.996 + 0.0774i)11-s + (−0.875 − 0.483i)12-s + (−0.993 + 0.116i)13-s + (−0.431 − 0.902i)14-s + (−0.790 − 0.612i)15-s + (−0.981 + 0.192i)16-s + (0.396 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000518 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.000518 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.000518 + 0.999i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.000518 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4517686022 + 0.4515342753i\)
\(L(\frac12)\) \(\approx\) \(0.4517686022 + 0.4515342753i\)
\(L(1)\) \(\approx\) \(0.6326833094 + 0.2126463078i\)
\(L(1)\) \(\approx\) \(0.6326833094 + 0.2126463078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.740 - 0.672i)T \)
3 \( 1 + (-0.565 + 0.824i)T \)
5 \( 1 + (-0.0581 + 0.998i)T \)
7 \( 1 + (0.925 + 0.378i)T \)
11 \( 1 + (0.996 + 0.0774i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (0.396 + 0.918i)T \)
19 \( 1 + (0.533 + 0.845i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.952 + 0.305i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
37 \( 1 + (-0.686 + 0.727i)T \)
41 \( 1 + (0.0968 - 0.995i)T \)
43 \( 1 + (-0.963 + 0.268i)T \)
47 \( 1 + (-0.211 + 0.977i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (-0.910 + 0.413i)T \)
71 \( 1 + (0.249 - 0.968i)T \)
73 \( 1 + (-0.910 - 0.413i)T \)
79 \( 1 + (-0.627 - 0.778i)T \)
83 \( 1 + (-0.135 - 0.990i)T \)
89 \( 1 + (0.996 - 0.0774i)T \)
97 \( 1 + (0.987 - 0.154i)T \)
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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

−27.61517958984130402940614784794, −26.7271446421794829505013321483, −25.05262206567704822751573607792, −24.71643747123510614013305266179, −23.923184509040601725416969743991, −23.13230217755054895550343612360, −21.76118035524644829262306537541, −20.03505859909525795971323040177, −19.74549208965873427968424061564, −18.27521899996371581187118142775, −17.47913318088601564695194557909, −16.889847830231134005472373148024, −15.947754303537723961051531834045, −14.40897095192466429032126883051, −13.64965378675935861459525204053, −12.07233986699439696134669195746, −11.414994803232374542029927713899, −9.91993262030343379688489881399, −8.72748738539560456176003457644, −7.71886347996614339624990805650, −6.91644947041919395257335263676, −5.45370926733554721081235503518, −4.72947393748783863579111011553, −1.85875178682650360535985063398, −0.77294644539073397112769537463, 1.80011594900818127168191703429, 3.3370392986926139989793019241, 4.40845946029601830350990651659, 6.049226471604979546203504805112, 7.40432647553014509932258892993, 8.66808954766211015828535487173, 9.9232401894275546723311215414, 10.53663803452205246657480909430, 11.750635046547621716501760497764, 12.07797772035321819074619656598, 14.29142116535727100463270927770, 14.98259811691456433176163982562, 16.35820934254697539793087117567, 17.34304102021384177584197778022, 18.00435528219215974783287635967, 19.08914498703113366707944064321, 20.16639010574903428982723733391, 21.27998896781814195636681146481, 22.01120892978571862671183953541, 22.58201167569281327195108130614, 24.11897032858876427484437179733, 25.45972771466384020289387014856, 26.42773457188985686921794701443, 27.35049797861860605133589428196, 27.61165046793771224102121991345

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