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Properties

Label 1-163-163.24-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.957 - 0.289i$
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  + (−0.627 − 0.778i)2-s + (0.533 − 0.845i)3-s + (−0.211 + 0.977i)4-s + (−0.686 − 0.727i)5-s + (−0.993 + 0.116i)6-s + (0.657 − 0.753i)7-s + (0.893 − 0.448i)8-s + (−0.431 − 0.902i)9-s + (−0.135 + 0.990i)10-s + (0.466 + 0.884i)11-s + (0.713 + 0.700i)12-s + (−0.0581 − 0.998i)13-s + (−0.999 − 0.0387i)14-s + (−0.981 + 0.192i)15-s + (−0.910 − 0.413i)16-s + (−0.835 − 0.549i)17-s + ⋯
L(s)  = 1  + (−0.627 − 0.778i)2-s + (0.533 − 0.845i)3-s + (−0.211 + 0.977i)4-s + (−0.686 − 0.727i)5-s + (−0.993 + 0.116i)6-s + (0.657 − 0.753i)7-s + (0.893 − 0.448i)8-s + (−0.431 − 0.902i)9-s + (−0.135 + 0.990i)10-s + (0.466 + 0.884i)11-s + (0.713 + 0.700i)12-s + (−0.0581 − 0.998i)13-s + (−0.999 − 0.0387i)14-s + (−0.981 + 0.192i)15-s + (−0.910 − 0.413i)16-s + (−0.835 − 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1198078430 - 0.8093565581i\)
\(L(\frac12)\) \(\approx\) \(0.1198078430 - 0.8093565581i\)
\(L(1)\) \(\approx\) \(0.5424082034 - 0.6245679630i\)
\(L(1)\) \(\approx\) \(0.5424082034 - 0.6245679630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.627 - 0.778i)T \)
3 \( 1 + (0.533 - 0.845i)T \)
5 \( 1 + (-0.686 - 0.727i)T \)
7 \( 1 + (0.657 - 0.753i)T \)
11 \( 1 + (0.466 + 0.884i)T \)
13 \( 1 + (-0.0581 - 0.998i)T \)
17 \( 1 + (-0.835 - 0.549i)T \)
19 \( 1 + (0.0193 + 0.999i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.360 - 0.932i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
37 \( 1 + (0.396 + 0.918i)T \)
41 \( 1 + (-0.211 - 0.977i)T \)
43 \( 1 + (-0.790 + 0.612i)T \)
47 \( 1 + (0.987 - 0.154i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.893 + 0.448i)T \)
67 \( 1 + (0.952 + 0.305i)T \)
71 \( 1 + (0.925 + 0.378i)T \)
73 \( 1 + (0.952 - 0.305i)T \)
79 \( 1 + (0.996 - 0.0774i)T \)
83 \( 1 + (0.323 + 0.946i)T \)
89 \( 1 + (0.466 - 0.884i)T \)
97 \( 1 + (-0.565 - 0.824i)T \)
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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

−27.782986841580161262738724567623, −27.02216920100942632564193532712, −26.399114607667709380390546349150, −25.544379374856381885517138556980, −24.37875507001712989109750640271, −23.65152713228251702760093570428, −22.123456525725224421124463192919, −21.65443512561266368822942213110, −19.99859814090580508601467804799, −19.26364772101384118600453348926, −18.42593401131291475860639635724, −17.15570891457015068937282968514, −16.06742784219355100718734111521, −15.26495561001823182618017304781, −14.623779279180970152049197159455, −13.69923403585927102218327002950, −11.33389416080552239419279344478, −11.00029785279877666881289523939, −9.36240721764328995709719049594, −8.723175273318825772286242686615, −7.69509562267688104084813893877, −6.41287462517534960612984152222, −5.0445742604780906933608795455, −3.831935311648663800222044575977, −2.17833489774132951969276770339, 0.82314205054305045360686249429, 2.004138956328056952640748714220, 3.57122375294470579340249670053, 4.67056332205267766088074924527, 6.9848846973708690703746375641, 7.88512387937439276689618058178, 8.56217115070958397678701035587, 9.81040327897200703478317585468, 11.16498701920581421827147033101, 12.204102280061897022541115489421, 12.8549647864296614218931767296, 13.98874134585420493990537336927, 15.30598702707792070733709716059, 16.83238165576014228081272705325, 17.58954186964722693682406592984, 18.526433704209830311561778904335, 19.665300086494302419554980971211, 20.44467809831320239886211702309, 20.60774083525302090345426957820, 22.55271448108702335359493037500, 23.39265877741170870471070155122, 24.67708658250752765321367866189, 25.24238264402149358854090842080, 26.67750127903086400026846774633, 27.243142777225629620227483874678

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