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Properties

Label 1-163-163.43-r0-0-0
Degree $1$
Conductor $163$
Sign $0.269 + 0.963i$
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  + (0.0968 + 0.995i)2-s + (−0.360 − 0.932i)3-s + (−0.981 + 0.192i)4-s + (−0.993 − 0.116i)5-s + (0.893 − 0.448i)6-s + (0.713 + 0.700i)7-s + (−0.286 − 0.957i)8-s + (−0.740 + 0.672i)9-s + (0.0193 − 0.999i)10-s + (0.987 + 0.154i)11-s + (0.533 + 0.845i)12-s + (0.973 − 0.230i)13-s + (−0.627 + 0.778i)14-s + (0.249 + 0.968i)15-s + (0.925 − 0.378i)16-s + (−0.686 + 0.727i)17-s + ⋯
L(s)  = 1  + (0.0968 + 0.995i)2-s + (−0.360 − 0.932i)3-s + (−0.981 + 0.192i)4-s + (−0.993 − 0.116i)5-s + (0.893 − 0.448i)6-s + (0.713 + 0.700i)7-s + (−0.286 − 0.957i)8-s + (−0.740 + 0.672i)9-s + (0.0193 − 0.999i)10-s + (0.987 + 0.154i)11-s + (0.533 + 0.845i)12-s + (0.973 − 0.230i)13-s + (−0.627 + 0.778i)14-s + (0.249 + 0.968i)15-s + (0.925 − 0.378i)16-s + (−0.686 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.269 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.269 + 0.963i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.269 + 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6701152157 + 0.5083749503i\)
\(L(\frac12)\) \(\approx\) \(0.6701152157 + 0.5083749503i\)
\(L(1)\) \(\approx\) \(0.7910566100 + 0.3164278234i\)
\(L(1)\) \(\approx\) \(0.7910566100 + 0.3164278234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.0968 + 0.995i)T \)
3 \( 1 + (-0.360 - 0.932i)T \)
5 \( 1 + (-0.993 - 0.116i)T \)
7 \( 1 + (0.713 + 0.700i)T \)
11 \( 1 + (0.987 + 0.154i)T \)
13 \( 1 + (0.973 - 0.230i)T \)
17 \( 1 + (-0.686 + 0.727i)T \)
19 \( 1 + (-0.431 + 0.902i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.813 + 0.581i)T \)
31 \( 1 + (0.396 + 0.918i)T \)
37 \( 1 + (-0.0581 - 0.998i)T \)
41 \( 1 + (-0.981 - 0.192i)T \)
43 \( 1 + (0.856 - 0.516i)T \)
47 \( 1 + (-0.910 - 0.413i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.657 - 0.753i)T \)
71 \( 1 + (-0.875 - 0.483i)T \)
73 \( 1 + (0.657 + 0.753i)T \)
79 \( 1 + (-0.211 + 0.977i)T \)
83 \( 1 + (-0.963 + 0.268i)T \)
89 \( 1 + (0.987 - 0.154i)T \)
97 \( 1 + (0.952 - 0.305i)T \)
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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

−27.586919893436378364118133780528, −26.95624015680343838303632458049, −26.21083163452085846886207368169, −24.245075472643003904236037890823, −23.22107420645441064788837004302, −22.66225787176934439391282220221, −21.60708337513765285745610094247, −20.59120417534601468643346267918, −20.03309813778361717549201195260, −18.91099156968558845568768563366, −17.65549230194591494651080200280, −16.7655318206957304302615441177, −15.467843807944899506444441979492, −14.51670727905815323839554789966, −13.45578105508691064308156853126, −11.79340985062055202374199172640, −11.313238828105614387515703315214, −10.58744039617648528223557816252, −9.18237967436694921881108181656, −8.335804509514245626825126375531, −6.52656326853697840638294780485, −4.624954677161561006020780742604, −4.26982793418802049567403759152, −3.04669543992610955027872268515, −0.875169809288392900854863168834, 1.38490323795305069688520837196, 3.6481688763402922975595838278, 4.97730685312879664912809791169, 6.161141999771183448288785400914, 7.095260101086044005687728576018, 8.32878423891143995728657321844, 8.70832816832524379235061263275, 10.93859443635339913175631814655, 12.03239907702515237162387366655, 12.78614118799255501599853377077, 14.09785934455585122986370233286, 15.0067104721462838866181588598, 15.995792831870498163534044148768, 17.1274240230212180918810867077, 17.943829669864505517146538150264, 18.875514468421175034805487899314, 19.718084033190138425708797065924, 21.37370005748555739812016124773, 22.61058801371237932181331649805, 23.2955424308459072340453009828, 24.096454650844287309628885150988, 24.90770351130716737600277376989, 25.5944555563071420660853629680, 27.14791820250306181728234782060, 27.721710350135558896125531677656

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