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Properties

Label 1-163-163.151-r0-0-0
Degree $1$
Conductor $163$
Sign $0.883 + 0.468i$
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  + (0.657 + 0.753i)2-s + (−0.211 − 0.977i)3-s + (−0.135 + 0.990i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)6-s + (0.856 − 0.516i)7-s + (−0.835 + 0.549i)8-s + (−0.910 + 0.413i)9-s + (0.466 + 0.884i)10-s + (−0.740 + 0.672i)11-s + (0.996 − 0.0774i)12-s + (0.893 − 0.448i)13-s + (0.952 + 0.305i)14-s + (0.0193 − 0.999i)15-s + (−0.963 − 0.268i)16-s + (−0.0581 − 0.998i)17-s + ⋯
L(s)  = 1  + (0.657 + 0.753i)2-s + (−0.211 − 0.977i)3-s + (−0.135 + 0.990i)4-s + (0.973 + 0.230i)5-s + (0.597 − 0.802i)6-s + (0.856 − 0.516i)7-s + (−0.835 + 0.549i)8-s + (−0.910 + 0.413i)9-s + (0.466 + 0.884i)10-s + (−0.740 + 0.672i)11-s + (0.996 − 0.0774i)12-s + (0.893 − 0.448i)13-s + (0.952 + 0.305i)14-s + (0.0193 − 0.999i)15-s + (−0.963 − 0.268i)16-s + (−0.0581 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.883 + 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643803983 + 0.4085031801i\)
\(L(\frac12)\) \(\approx\) \(1.643803983 + 0.4085031801i\)
\(L(1)\) \(\approx\) \(1.504038885 + 0.2950030525i\)
\(L(1)\) \(\approx\) \(1.504038885 + 0.2950030525i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.657 + 0.753i)T \)
3 \( 1 + (-0.211 - 0.977i)T \)
5 \( 1 + (0.973 + 0.230i)T \)
7 \( 1 + (0.856 - 0.516i)T \)
11 \( 1 + (-0.740 + 0.672i)T \)
13 \( 1 + (0.893 - 0.448i)T \)
17 \( 1 + (-0.0581 - 0.998i)T \)
19 \( 1 + (0.987 - 0.154i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.981 - 0.192i)T \)
31 \( 1 + (-0.686 + 0.727i)T \)
37 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (-0.135 - 0.990i)T \)
43 \( 1 + (0.533 + 0.845i)T \)
47 \( 1 + (0.323 - 0.946i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.835 - 0.549i)T \)
67 \( 1 + (-0.790 + 0.612i)T \)
71 \( 1 + (-0.999 + 0.0387i)T \)
73 \( 1 + (-0.790 - 0.612i)T \)
79 \( 1 + (0.813 - 0.581i)T \)
83 \( 1 + (-0.875 - 0.483i)T \)
89 \( 1 + (-0.740 - 0.672i)T \)
97 \( 1 + (0.0968 + 0.995i)T \)
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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

−28.138347651138769948583374839216, −26.86507283675501702306723487883, −25.85233907556987052481251110252, −24.46354829353214358699255938387, −23.72201974183770082631806520902, −22.34255859721898351490328871551, −21.729123896420003917714636601462, −20.83787077326945830042233367624, −20.57179108829731851953235295157, −18.77343550427500582510862388634, −17.96291158018381742590437259392, −16.65352073567644273207187112113, −15.52628196443624372353513545756, −14.49533465676467166565010024685, −13.70045150035466505200381988156, −12.48897483658950155528000631149, −11.161166801502677454850831051488, −10.6522984265546228728629714358, −9.41120779234250158561416488914, −8.55158055557753061690023431601, −5.997649730112263852147878347799, −5.4762360786297852884315776577, −4.36037813457427400967824137534, −3.00485564453632276740274292637, −1.627247525572326865236573400686, 1.69420791180339258798236680540, 3.10862211233653219485200162474, 5.06724568272126522495671844842, 5.69840448836154873433186274971, 7.110250342870902955099650617436, 7.636148154725669166549610764726, 9.03328086102632377819217521270, 10.76317009817933693123237855395, 11.86142135973429630050011528250, 13.189157327375024409695964255946, 13.643315565791390737494472263108, 14.52726613393606778479185143058, 15.84723400055370311172026792860, 17.14288335652325086891242841092, 17.90871159159216593237438269191, 18.320240920998872493297459008065, 20.362632222478837457807558949, 20.97784659303391582886041236447, 22.34614185350576066483735749873, 23.10808775142111870795268022152, 23.96744946616214497336627425146, 24.852966792999154140388631310972, 25.57490361491662391339498415609, 26.40809741355023023722769493671, 27.82307005886877866849778596334

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