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Properties

Label 1-163-163.121-r0-0-0
Degree $1$
Conductor $163$
Sign $0.860 + 0.508i$
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  + (−0.875 − 0.483i)2-s + (−0.790 − 0.612i)3-s + (0.533 + 0.845i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (−0.627 + 0.778i)7-s + (−0.0581 − 0.998i)8-s + (0.249 + 0.968i)9-s + (−0.211 + 0.977i)10-s + (−0.135 + 0.990i)11-s + (0.0968 − 0.995i)12-s + (−0.835 − 0.549i)13-s + (0.925 − 0.378i)14-s + (−0.360 + 0.932i)15-s + (−0.431 + 0.902i)16-s + (0.893 + 0.448i)17-s + ⋯
L(s)  = 1  + (−0.875 − 0.483i)2-s + (−0.790 − 0.612i)3-s + (0.533 + 0.845i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (−0.627 + 0.778i)7-s + (−0.0581 − 0.998i)8-s + (0.249 + 0.968i)9-s + (−0.211 + 0.977i)10-s + (−0.135 + 0.990i)11-s + (0.0968 − 0.995i)12-s + (−0.835 − 0.549i)13-s + (0.925 − 0.378i)14-s + (−0.360 + 0.932i)15-s + (−0.431 + 0.902i)16-s + (0.893 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.860 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3301185840 + 0.09028286379i\)
\(L(\frac12)\) \(\approx\) \(0.3301185840 + 0.09028286379i\)
\(L(1)\) \(\approx\) \(0.4452017775 - 0.09101485895i\)
\(L(1)\) \(\approx\) \(0.4452017775 - 0.09101485895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.875 - 0.483i)T \)
3 \( 1 + (-0.790 - 0.612i)T \)
5 \( 1 + (-0.286 - 0.957i)T \)
7 \( 1 + (-0.627 + 0.778i)T \)
11 \( 1 + (-0.135 + 0.990i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.893 + 0.448i)T \)
19 \( 1 + (-0.981 - 0.192i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.856 + 0.516i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
37 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + (0.533 - 0.845i)T \)
43 \( 1 + (0.952 + 0.305i)T \)
47 \( 1 + (0.0193 + 0.999i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (-0.999 - 0.0387i)T \)
71 \( 1 + (-0.740 + 0.672i)T \)
73 \( 1 + (-0.999 + 0.0387i)T \)
79 \( 1 + (0.713 + 0.700i)T \)
83 \( 1 + (0.987 + 0.154i)T \)
89 \( 1 + (-0.135 - 0.990i)T \)
97 \( 1 + (-0.963 + 0.268i)T \)
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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

−27.340156984538422909384360878231, −26.63645807884973571563716273755, −26.28865334082227881496731166522, −24.871538536074385470340595314141, −23.51726638430792296284947501826, −23.11685990708174119710594500335, −21.92969303802930103434837996810, −20.81068771550795514766073830449, −19.359811325106997982847595598675, −18.837411691366587104066622981922, −17.59543003126443377245372912497, −16.65769549599740162785768361294, −16.099491143199573073997790578087, −14.9120802487464006317903977870, −14.04126546265050955530405065932, −12.093882667976131413378066892354, −10.96097300278761804175479083278, −10.34389218437112981062298064175, −9.45453730757504624079364486260, −7.87233548294347401976783738593, −6.70529188166707913885289953358, −6.06461867343314301957899265599, −4.42387203981763949809172340539, −2.8861194074288524668199678451, −0.45526882595448753410840320636, 1.291494394652057701959707840443, 2.65694030124609781837731932611, 4.5945902858427175409510947275, 5.93736272852634503344424604550, 7.29070995834985956658409386800, 8.21128571498739350338771529635, 9.466712772091329771379054858666, 10.42043372348511649062985813568, 11.904828715060414091661362883221, 12.41803324712474784858180737569, 13.026002530777811238367758186501, 15.30185070996724577059494530790, 16.223330805347462724603747828544, 17.241490910267382130996669526935, 17.78456985873790669813730542790, 19.2246062255101056590911138601, 19.50540634879715112388515939467, 20.89026133917599823708957940421, 21.86993728225754243907614601881, 22.96241291474035071670479085614, 24.05945125943364297784027620619, 25.19191845397317444962207947372, 25.59210761407379225887564540629, 27.46150855232518900885661557778, 27.851944554742386050751628281827

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