Properties

Label 1-163-163.47-r0-0-0
Degree $1$
Conductor $163$
Sign $0.758 - 0.651i$
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  + (0.466 + 0.884i)2-s + (−0.963 + 0.268i)3-s + (−0.565 + 0.824i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (−0.740 − 0.672i)7-s + (−0.993 − 0.116i)8-s + (0.856 − 0.516i)9-s + (0.0968 − 0.995i)10-s + (0.713 + 0.700i)11-s + (0.323 − 0.946i)12-s + (0.396 − 0.918i)13-s + (0.249 − 0.968i)14-s + (0.952 + 0.305i)15-s + (−0.360 − 0.932i)16-s + (0.597 − 0.802i)17-s + ⋯
L(s)  = 1  + (0.466 + 0.884i)2-s + (−0.963 + 0.268i)3-s + (−0.565 + 0.824i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (−0.740 − 0.672i)7-s + (−0.993 − 0.116i)8-s + (0.856 − 0.516i)9-s + (0.0968 − 0.995i)10-s + (0.713 + 0.700i)11-s + (0.323 − 0.946i)12-s + (0.396 − 0.918i)13-s + (0.249 − 0.968i)14-s + (0.952 + 0.305i)15-s + (−0.360 − 0.932i)16-s + (0.597 − 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.758 - 0.651i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4704408973 - 0.1743822324i\)
\(L(\frac12)\) \(\approx\) \(0.4704408973 - 0.1743822324i\)
\(L(1)\) \(\approx\) \(0.6550760729 + 0.1508074042i\)
\(L(1)\) \(\approx\) \(0.6550760729 + 0.1508074042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.466 + 0.884i)T \)
3 \( 1 + (-0.963 + 0.268i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.740 - 0.672i)T \)
11 \( 1 + (0.713 + 0.700i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (0.597 - 0.802i)T \)
19 \( 1 + (-0.790 - 0.612i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.999 + 0.0387i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
37 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (-0.565 - 0.824i)T \)
43 \( 1 + (-0.910 - 0.413i)T \)
47 \( 1 + (0.533 - 0.845i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (-0.431 + 0.902i)T \)
71 \( 1 + (0.813 - 0.581i)T \)
73 \( 1 + (-0.431 - 0.902i)T \)
79 \( 1 + (-0.875 + 0.483i)T \)
83 \( 1 + (-0.211 + 0.977i)T \)
89 \( 1 + (0.713 - 0.700i)T \)
97 \( 1 + (0.0193 - 0.999i)T \)
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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

−28.00728797275379099839762388895, −27.36811907793463994931504218803, −26.1022783001912396229838857246, −24.51564325970409613560074451909, −23.57299124894316696397736951530, −22.94529681654022177347099199055, −21.96810063098715818826943577122, −21.47890989901562257817993292023, −19.75909563365173977019693276656, −18.80588659374946523372831764957, −18.674103478930474039221129023862, −16.95957908259067874328282929620, −15.88756002266562631993005614079, −14.784080304724467898424071990590, −13.55067362741222567677161205192, −12.29504462163717505397549435710, −11.80751047762712531259130404859, −10.88542887271325843284430071212, −9.82337286413585175306377274568, −8.36786334595899580447833374160, −6.50211659888579053976496071186, −5.95467108965508985213875523253, −4.29536612844932679466754602731, −3.34751234199967886008093269987, −1.619688953750423375375263971589, 0.42192266204453942181160043380, 3.636477143280445389700190787611, 4.37793445827991388342695704178, 5.54177327397230480983785215042, 6.74236076276342519361364283527, 7.57241947125086024820163766247, 9.03230747555372327441917711314, 10.25973834655891003122375914513, 11.76696197047532897945094890393, 12.50819902035784490169936459559, 13.44233001988273516824788285096, 15.035573010846554336237095482305, 15.7840458445302497977755270637, 16.65101424195647860773384283023, 17.26354203403160293181710927764, 18.46393636691632545003447616450, 19.947715349919205827437128722313, 20.93076480249428876671211500563, 22.40685042443107730220028761323, 22.83277634503181517726553143124, 23.56363979411093776452782922812, 24.49885994765468697684251618231, 25.613458752786830869944662034314, 26.69710179267233922939924354076, 27.64845814204783874754600956734

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