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Properties

Label 1-163-163.46-r0-0-0
Degree $1$
Conductor $163$
Sign $0.144 + 0.989i$
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  + (0.925 + 0.378i)2-s + (0.0968 + 0.995i)3-s + (0.713 + 0.700i)4-s + (0.893 − 0.448i)5-s + (−0.286 + 0.957i)6-s + (−0.999 − 0.0387i)7-s + (0.396 + 0.918i)8-s + (−0.981 + 0.192i)9-s + (0.996 − 0.0774i)10-s + (0.813 − 0.581i)11-s + (−0.627 + 0.778i)12-s + (0.597 + 0.802i)13-s + (−0.910 − 0.413i)14-s + (0.533 + 0.845i)15-s + (0.0193 + 0.999i)16-s + (−0.993 − 0.116i)17-s + ⋯
L(s)  = 1  + (0.925 + 0.378i)2-s + (0.0968 + 0.995i)3-s + (0.713 + 0.700i)4-s + (0.893 − 0.448i)5-s + (−0.286 + 0.957i)6-s + (−0.999 − 0.0387i)7-s + (0.396 + 0.918i)8-s + (−0.981 + 0.192i)9-s + (0.996 − 0.0774i)10-s + (0.813 − 0.581i)11-s + (−0.627 + 0.778i)12-s + (0.597 + 0.802i)13-s + (−0.910 − 0.413i)14-s + (0.533 + 0.845i)15-s + (0.0193 + 0.999i)16-s + (−0.993 − 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.144 + 0.989i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.144 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.534494405 + 1.326169274i\)
\(L(\frac12)\) \(\approx\) \(1.534494405 + 1.326169274i\)
\(L(1)\) \(\approx\) \(1.589588270 + 0.8665092182i\)
\(L(1)\) \(\approx\) \(1.589588270 + 0.8665092182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.925 + 0.378i)T \)
3 \( 1 + (0.0968 + 0.995i)T \)
5 \( 1 + (0.893 - 0.448i)T \)
7 \( 1 + (-0.999 - 0.0387i)T \)
11 \( 1 + (0.813 - 0.581i)T \)
13 \( 1 + (0.597 + 0.802i)T \)
17 \( 1 + (-0.993 - 0.116i)T \)
19 \( 1 + (-0.211 - 0.977i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.790 - 0.612i)T \)
31 \( 1 + (-0.0581 + 0.998i)T \)
37 \( 1 + (0.973 + 0.230i)T \)
41 \( 1 + (0.713 - 0.700i)T \)
43 \( 1 + (-0.565 + 0.824i)T \)
47 \( 1 + (-0.135 - 0.990i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.396 - 0.918i)T \)
67 \( 1 + (-0.963 - 0.268i)T \)
71 \( 1 + (-0.431 - 0.902i)T \)
73 \( 1 + (-0.963 + 0.268i)T \)
79 \( 1 + (0.657 - 0.753i)T \)
83 \( 1 + (0.466 + 0.884i)T \)
89 \( 1 + (0.813 + 0.581i)T \)
97 \( 1 + (0.323 + 0.946i)T \)
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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

−28.00046248660725118343680090989, −26.03857984769771612058142186500, −25.30595807965109814505622328625, −24.720506294637881133734956961956, −23.44214835526343404200426611238, −22.53893091594716102222306727612, −22.08152645971063151125889760363, −20.52253311768276552301173630870, −19.840688296212146585830026013525, −18.78450725464788150203367335913, −17.87797494645664456796669817635, −16.612431390170297073460668079034, −15.14827767412406314257638552659, −14.23658641585900589457166490873, −13.225373619130482053566391570220, −12.7631021634881106772510209597, −11.54724832115757521455981276191, −10.32748617919292557444708258206, −9.2397810303997720324494930718, −7.37675947501519380446277716100, −6.24798875103286124898585507585, −5.87868936912587695709473691748, −3.82621885903761710400887281738, −2.62322637026152365698453027040, −1.56819658729820886827965642893, 2.353425719219699086540935080503, 3.68519297541452733183100504508, 4.61993173963938652186968429430, 5.95826873538495647475672154713, 6.5886575100386452912092846463, 8.67121523553544315978112068684, 9.37458307742837849791030189107, 10.759499919159410914223391185764, 11.84310664594788542950569075931, 13.2991512845377295780328998782, 13.82032532298808236653522885649, 14.97705267147817444449970980808, 16.21554751593509115829687379859, 16.506132507758900459110021082550, 17.659879076207255930825883315061, 19.5972429238056210349700358042, 20.38412292748071360545693468161, 21.57629955609191873402992984892, 21.91674048519795899693151276206, 22.847416577181362775972689189924, 24.12591942155434360640159152434, 25.077167605404082992468212041074, 26.015162761489229377451780887442, 26.51723442751399348861985312873, 28.21160909421635418105345551606

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