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Properties

Label 1-163-163.156-r0-0-0
Degree $1$
Conductor $163$
Sign $0.999 + 0.0276i$
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  + (0.952 − 0.305i)2-s + (0.996 + 0.0774i)3-s + (0.813 − 0.581i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)6-s + (−0.790 + 0.612i)7-s + (0.597 − 0.802i)8-s + (0.987 + 0.154i)9-s + (0.249 + 0.968i)10-s + (−0.431 − 0.902i)11-s + (0.856 − 0.516i)12-s + (−0.993 + 0.116i)13-s + (−0.565 + 0.824i)14-s + (−0.135 + 0.990i)15-s + (0.323 − 0.946i)16-s + (0.396 + 0.918i)17-s + ⋯
L(s)  = 1  + (0.952 − 0.305i)2-s + (0.996 + 0.0774i)3-s + (0.813 − 0.581i)4-s + (−0.0581 + 0.998i)5-s + (0.973 − 0.230i)6-s + (−0.790 + 0.612i)7-s + (0.597 − 0.802i)8-s + (0.987 + 0.154i)9-s + (0.249 + 0.968i)10-s + (−0.431 − 0.902i)11-s + (0.856 − 0.516i)12-s + (−0.993 + 0.116i)13-s + (−0.565 + 0.824i)14-s + (−0.135 + 0.990i)15-s + (0.323 − 0.946i)16-s + (0.396 + 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.999 + 0.0276i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.999 + 0.0276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.375807487 - 0.03287175069i\)
\(L(\frac12)\) \(\approx\) \(2.375807487 - 0.03287175069i\)
\(L(1)\) \(\approx\) \(2.087882082 - 0.07096902742i\)
\(L(1)\) \(\approx\) \(2.087882082 - 0.07096902742i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.952 - 0.305i)T \)
3 \( 1 + (0.996 + 0.0774i)T \)
5 \( 1 + (-0.0581 + 0.998i)T \)
7 \( 1 + (-0.790 + 0.612i)T \)
11 \( 1 + (-0.431 - 0.902i)T \)
13 \( 1 + (-0.993 + 0.116i)T \)
17 \( 1 + (0.396 + 0.918i)T \)
19 \( 1 + (0.466 - 0.884i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.211 - 0.977i)T \)
31 \( 1 + (-0.835 - 0.549i)T \)
37 \( 1 + (-0.686 + 0.727i)T \)
41 \( 1 + (0.813 + 0.581i)T \)
43 \( 1 + (0.713 + 0.700i)T \)
47 \( 1 + (-0.740 - 0.672i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.597 + 0.802i)T \)
67 \( 1 + (0.0968 - 0.995i)T \)
71 \( 1 + (-0.963 + 0.268i)T \)
73 \( 1 + (0.0968 + 0.995i)T \)
79 \( 1 + (-0.360 + 0.932i)T \)
83 \( 1 + (0.925 + 0.378i)T \)
89 \( 1 + (-0.431 + 0.902i)T \)
97 \( 1 + (-0.627 - 0.778i)T \)
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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

−27.61846984864823811804604480800, −26.419686904466958647010279807235, −25.53720930091997450651112800875, −24.85464592151714715281600805697, −23.94364351069880736222544286012, −23.03008007014595297137196430875, −21.9081228249872981628608474189, −20.60229972641760165594027784736, −20.31402278878702144080691712782, −19.38652147898355382936913686294, −17.68313526664596154729717189158, −16.32632855353713515274367263705, −15.85386349163184951658929134006, −14.52788397590710536330744227877, −13.76620935672465384231121213288, −12.63207229676694761756895284381, −12.304795603450164636409101658450, −10.21893250616891687196272222312, −9.244573926112735299088116032233, −7.689326740616819686379751609547, −7.21952186240911268998545901562, −5.43261520515465941357256257784, −4.329655847802994438188303011324, −3.28783924009367641369845301869, −1.89780064742174923920670313559, 2.29907525722399158745253283054, 2.976083924979495209184376236013, 3.98417955429437860927754497370, 5.67628967781559945355117511995, 6.77440363773992567915407303316, 7.92066544697358941731563785455, 9.571401877780521441473536629, 10.385608849439979417293280651620, 11.67105231004224497854216179112, 12.866291664937450283714770255822, 13.73251170106064463668115170608, 14.7068085097444671492210663723, 15.35911275657316202465381113713, 16.32826343959832253599717196397, 18.37874512371616790935444128198, 19.29306659828347172104228735240, 19.7081673899609899715405237901, 21.20186520531830490552752209663, 21.86888194011613035222885864799, 22.54249281079142798783728366158, 23.974557396822014174287001547973, 24.66379204608433116368177111413, 25.95776963142090474092877943886, 26.30880531451067612874528252581, 27.78594834571205971141023502359

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