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Properties

Label 1-163-163.161-r0-0-0
Degree $1$
Conductor $163$
Sign $0.994 - 0.108i$
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  + (−0.999 − 0.0387i)2-s + (0.713 + 0.700i)3-s + (0.996 + 0.0774i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (0.952 − 0.305i)7-s + (−0.993 − 0.116i)8-s + (0.0193 + 0.999i)9-s + (0.813 + 0.581i)10-s + (0.249 − 0.968i)11-s + (0.657 + 0.753i)12-s + (0.396 − 0.918i)13-s + (−0.963 + 0.268i)14-s + (−0.211 − 0.977i)15-s + (0.987 + 0.154i)16-s + (0.597 − 0.802i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0387i)2-s + (0.713 + 0.700i)3-s + (0.996 + 0.0774i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)6-s + (0.952 − 0.305i)7-s + (−0.993 − 0.116i)8-s + (0.0193 + 0.999i)9-s + (0.813 + 0.581i)10-s + (0.249 − 0.968i)11-s + (0.657 + 0.753i)12-s + (0.396 − 0.918i)13-s + (−0.963 + 0.268i)14-s + (−0.211 − 0.977i)15-s + (0.987 + 0.154i)16-s + (0.597 − 0.802i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9028711267 - 0.04899914653i\)
\(L(\frac12)\) \(\approx\) \(0.9028711267 - 0.04899914653i\)
\(L(1)\) \(\approx\) \(0.8673268268 + 0.01177050311i\)
\(L(1)\) \(\approx\) \(0.8673268268 + 0.01177050311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.999 - 0.0387i)T \)
3 \( 1 + (0.713 + 0.700i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (0.952 - 0.305i)T \)
11 \( 1 + (0.249 - 0.968i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (0.597 - 0.802i)T \)
19 \( 1 + (-0.135 + 0.990i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.533 + 0.845i)T \)
31 \( 1 + (0.893 - 0.448i)T \)
37 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (0.996 - 0.0774i)T \)
43 \( 1 + (0.0968 + 0.995i)T \)
47 \( 1 + (0.466 + 0.884i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (-0.565 - 0.824i)T \)
71 \( 1 + (-0.910 - 0.413i)T \)
73 \( 1 + (-0.565 + 0.824i)T \)
79 \( 1 + (0.856 + 0.516i)T \)
83 \( 1 + (-0.740 - 0.672i)T \)
89 \( 1 + (0.249 + 0.968i)T \)
97 \( 1 + (-0.875 + 0.483i)T \)
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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

−27.7934979792880349462678887134, −26.5698394480308344280565121491, −26.01569116814829478430812397044, −25.03640600153330056641253914207, −23.98840551970494746847105035076, −23.436783234835484755783046778847, −21.56946212421606254615422952952, −20.55171538864279083006111602400, −19.62508378776071482868945140048, −18.93967552890445725960922450676, −18.05474165123623100925475625795, −17.26343912536610745886614022304, −15.59141771887733037198456170175, −14.99241330582887936265801669293, −13.99660826878390123745636563465, −12.12791950395347109321204820311, −11.69146313721441666254942415301, −10.32243805184632689850355212126, −8.96772857649565348372388080238, −8.10986421746567522628285488664, −7.311741916617434723491055888158, −6.34598763202410000060340814641, −4.12782163730253898727300785911, −2.57177309732352804832182464474, −1.496352675665215152325538602115, 1.159969320502589308915685953150, 2.96803472852374051705562629589, 4.124971046308637274612885152905, 5.635451308763580966429170925447, 7.70328351513608777349322049692, 8.13497040308754699617790581178, 9.02810367945810179460107920681, 10.35631507146161186515826695422, 11.15928549611533420890421589048, 12.26133041218123490266391319360, 13.98741366467320518332809721011, 14.96886824579940019241892245198, 16.10293216000058818736885481965, 16.52028552099316159329976818863, 17.91992513295510580087378847210, 19.04390738299955450262381072469, 19.93305064093754889491525136084, 20.68757004948815781919751775666, 21.29304906198850454532265502246, 22.928446507965291648268595506254, 24.3452716295162018352137504697, 24.836614932132187843519577495, 26.044832969531087822983274771857, 27.133215750955583673767725799060, 27.39864424430656847576636026626

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