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Properties

Label 1-163-163.9-r0-0-0
Degree $1$
Conductor $163$
Sign $0.998 + 0.0525i$
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  + (0.0193 + 0.999i)2-s + (0.925 − 0.378i)3-s + (−0.999 + 0.0387i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (0.987 + 0.154i)7-s + (−0.0581 − 0.998i)8-s + (0.713 − 0.700i)9-s + (0.952 − 0.305i)10-s + (−0.790 − 0.612i)11-s + (−0.910 + 0.413i)12-s + (−0.835 − 0.549i)13-s + (−0.135 + 0.990i)14-s + (−0.627 − 0.778i)15-s + (0.996 − 0.0774i)16-s + (0.893 + 0.448i)17-s + ⋯
L(s)  = 1  + (0.0193 + 0.999i)2-s + (0.925 − 0.378i)3-s + (−0.999 + 0.0387i)4-s + (−0.286 − 0.957i)5-s + (0.396 + 0.918i)6-s + (0.987 + 0.154i)7-s + (−0.0581 − 0.998i)8-s + (0.713 − 0.700i)9-s + (0.952 − 0.305i)10-s + (−0.790 − 0.612i)11-s + (−0.910 + 0.413i)12-s + (−0.835 − 0.549i)13-s + (−0.135 + 0.990i)14-s + (−0.627 − 0.778i)15-s + (0.996 − 0.0774i)16-s + (0.893 + 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.998 + 0.0525i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.998 + 0.0525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.373750027 + 0.03612897029i\)
\(L(\frac12)\) \(\approx\) \(1.373750027 + 0.03612897029i\)
\(L(1)\) \(\approx\) \(1.264255692 + 0.1610376861i\)
\(L(1)\) \(\approx\) \(1.264255692 + 0.1610376861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.0193 + 0.999i)T \)
3 \( 1 + (0.925 - 0.378i)T \)
5 \( 1 + (-0.286 - 0.957i)T \)
7 \( 1 + (0.987 + 0.154i)T \)
11 \( 1 + (-0.790 - 0.612i)T \)
13 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.893 + 0.448i)T \)
19 \( 1 + (0.657 - 0.753i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.875 + 0.483i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
37 \( 1 + (0.597 + 0.802i)T \)
41 \( 1 + (-0.999 - 0.0387i)T \)
43 \( 1 + (-0.740 + 0.672i)T \)
47 \( 1 + (0.856 - 0.516i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.0581 + 0.998i)T \)
67 \( 1 + (0.466 + 0.884i)T \)
71 \( 1 + (-0.211 - 0.977i)T \)
73 \( 1 + (0.466 - 0.884i)T \)
79 \( 1 + (-0.963 + 0.268i)T \)
83 \( 1 + (-0.360 - 0.932i)T \)
89 \( 1 + (-0.790 + 0.612i)T \)
97 \( 1 + (0.249 - 0.968i)T \)
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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

−27.54009084675471302340104972498, −26.76632193111541872912925111810, −26.32890332394961822350878582588, −24.941533195163055886810423154193, −23.63417251513091165548300510834, −22.622664290445321437391580183314, −21.62777563212923975102963970587, −20.764374236306538956093324223018, −20.14503485432713149376002186836, −18.7590841120801209079886800733, −18.51883968273708054328150199369, −17.04920676986257900574576712740, −15.368499284334515535818590225414, −14.408715568101718302356618354478, −13.99148845268334642167546778766, −12.476186488873103342052704950337, −11.3779055527984698542834984992, −10.304511861803965773806810109258, −9.65603608135476948891617023931, −8.118533300002527827145641008871, −7.45269368479647860524122385721, −5.09802616969649853460836853934, −4.09326836795245463044870706090, −2.856704832555194873963167923448, −1.95613900336859748763210551925, 1.21687141129730420623083811367, 3.21769545158571155928381175204, 4.71900232442887219974590211289, 5.55895879321038054570576806779, 7.43472086343567782028366689672, 7.99836158904505348888810122993, 8.79950817697679842517294757187, 9.93034486665490830862848466668, 11.899441392023014985455304464810, 13.02462534293406588787634310579, 13.76163095401445134659357464539, 14.936402079650579708160551332814, 15.559166439129162316584628696591, 16.815605770419550995056809690593, 17.793763378818877243256068109382, 18.7760827300177419353771282940, 19.84027004006518312713822318172, 20.938067116266096688996983076751, 21.784985099747914015931799289354, 23.56838500397016888698931005426, 24.03055461638114963477471277922, 24.75875426534256215819209206127, 25.579414295223486729399312567029, 26.75566112950985236974581041207, 27.35813841797987221341366107454

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