Properties

Label 1-163-163.55-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: $\chi_{163} (55, \cdot )$
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

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

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