Properties

Label 1-163-163.58-r0-0-0
Degree $1$
Conductor $163$
Sign $0.997 - 0.0682i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + 6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + 6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.997 - 0.0682i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.997 - 0.0682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8062101647 + 0.02754660213i\)
\(L(\frac12)\) \(\approx\) \(0.8062101647 + 0.02754660213i\)
\(L(1)\) \(\approx\) \(0.7917002217 - 0.07566116804i\)
\(L(1)\) \(\approx\) \(0.7917002217 - 0.07566116804i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−28.062276915144401823588207770169, −26.49524705259149751379937551192, −25.43662573953632946792190906316, −25.11303884503878679004259904190, −24.03378474969735422877868535895, −23.16744809376420345556558293946, −22.16382594685919214078919061525, −21.0735186060872807172610310503, −19.22622567882228106018467972348, −18.69447345181167078631270287650, −17.91615806902958819817389944015, −16.91945890411194847906765031202, −16.123000070433092827196737717355, −14.84682135549748561671524902162, −13.46037000545413638810893256941, −13.14132639678603747732662068693, −11.41909501743557287547422314435, −10.27028850510026310924428604540, −9.00824875205717638116535135642, −8.13950832178129417619497256196, −6.65375445185311845573455482306, −5.96287117462564529619032440283, −5.23862781578164955322436600771, −2.63394123067220326486547473918, −1.07698211400040543456137616392, 1.2799052282759393868792856550, 3.04379116024399915177673002722, 4.162805165228431770498577351586, 5.45221148877761848812317001722, 6.89989558057537128094236711326, 8.54988542562660878314400994884, 9.85264552495449287059685377212, 10.16212934132270188532976191536, 11.12452013721289632040708648694, 12.540141071552080607726872494741, 13.39041440717362574269974831320, 14.66330671220986007404277522485, 16.30260573817613060891635035131, 16.91577809093266140593113493744, 17.811790851927330221987467693933, 18.76817752568995990989944579265, 20.3012449198903141790085871980, 20.85722016001420309268426919838, 21.557025822992825361988034368133, 22.86291337714497083546034830498, 23.196824237564775206693472866593, 25.48050391386206535170450062054, 25.90200673444030067756538675757, 26.89694764506618722904463329591, 27.91030172061397147826664130877

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