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Properties

Label 1-163-163.126-r0-0-0
Degree $1$
Conductor $163$
Sign $0.999 + 0.0183i$
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  + (−0.286 − 0.957i)2-s + (0.893 + 0.448i)3-s + (−0.835 + 0.549i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.686 + 0.727i)7-s + (0.766 + 0.642i)8-s + (0.597 + 0.802i)9-s + (−0.0581 + 0.998i)10-s + (0.893 + 0.448i)11-s + (−0.993 + 0.116i)12-s + (0.766 − 0.642i)13-s + (0.893 + 0.448i)14-s + (−0.686 − 0.727i)15-s + (0.396 − 0.918i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.286 − 0.957i)2-s + (0.893 + 0.448i)3-s + (−0.835 + 0.549i)4-s + (−0.939 − 0.342i)5-s + (0.173 − 0.984i)6-s + (−0.686 + 0.727i)7-s + (0.766 + 0.642i)8-s + (0.597 + 0.802i)9-s + (−0.0581 + 0.998i)10-s + (0.893 + 0.448i)11-s + (−0.993 + 0.116i)12-s + (0.766 − 0.642i)13-s + (0.893 + 0.448i)14-s + (−0.686 − 0.727i)15-s + (0.396 − 0.918i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.999 + 0.0183i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.999 + 0.0183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.030891951 + 0.009434608112i\)
\(L(\frac12)\) \(\approx\) \(1.030891951 + 0.009434608112i\)
\(L(1)\) \(\approx\) \(0.9876863609 - 0.1338689674i\)
\(L(1)\) \(\approx\) \(0.9876863609 - 0.1338689674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.286 - 0.957i)T \)
3 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (-0.686 + 0.727i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.973 - 0.230i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.286 + 0.957i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.173 + 0.984i)T \)
41 \( 1 + (-0.835 - 0.549i)T \)
43 \( 1 + (-0.0581 - 0.998i)T \)
47 \( 1 + (-0.286 - 0.957i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (-0.0581 - 0.998i)T \)
73 \( 1 + (-0.835 + 0.549i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (-0.686 + 0.727i)T \)
89 \( 1 + (0.893 - 0.448i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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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

−27.18291425685134801168257432611, −26.65474087469114991124457567690, −25.87840574306630113646602004692, −24.90753064196201512357991162899, −23.98272664140628982679625781279, −23.18240874833845503265299741219, −22.32630430302663911001064895532, −20.55356292241829038002748625758, −19.54380047376761073084544914170, −18.946441263465819720689801698058, −18.069855178248893665847092203037, −16.40874104481975408187111450, −16.03787136127211657571054845001, −14.55835074846003157938449375932, −14.107633883134042655080502588224, −12.99375505455087243506427979303, −11.57571404400840705163396472887, −9.97701290719702783764899957805, −9.00072613553897453859638130903, −7.91957286515263209596491023279, −7.10265517985765430162817078620, −6.25221818587944550064720304624, −4.147144046142579845779810200128, −3.402593622865139917036627481567, −1.02333567958415474735058039062, 1.59088370260452680352708783487, 3.32948896124514637250689389049, 3.70600449113649380635827647375, 5.243660646381069699295023478332, 7.44755123585422472994542759601, 8.51822139779300667473682780277, 9.24660070407366338440336172626, 10.24091505566372190354995244100, 11.59359272481871460249635909956, 12.48318910273479828725986410808, 13.449238687155336766759724634257, 14.78825987503745108005257212348, 15.753316353160414346259950192324, 16.75627558196919537532120206495, 18.351448505611668618780708232943, 19.16848431875167920347246587063, 20.01945793568355187455040717371, 20.49411176321436134650484219005, 21.85518300306518246877167940543, 22.44879597938202116503904934240, 23.73216854113962699142505638636, 25.28995706595353390196779876005, 25.829802535274458941627296049048, 27.07508765637632551618273451758, 27.79854826098132147971002528395

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