Properties

Degree 1
Conductor 163
Sign $0.144 - 0.989i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $0.144 - 0.989i$
motivic weight  =  \(0\)
character  :  $\chi_{163} (39, \cdot )$
Sato-Tate  :  $\mu(81)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 163,\ (0:\ ),\ 0.144 - 0.989i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.534494405 - 1.326169274i$
$L(\frac12,\chi)$  $\approx$  $1.534494405 - 1.326169274i$
$L(\chi,1)$  $\approx$  1.589588270 - 0.8665092182i
$L(1,\chi)$  $\approx$  1.589588270 - 0.8665092182i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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