Properties

Degree 1
Conductor 163
Sign $0.557 + 0.829i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.987 + 0.154i)2-s + (−0.999 + 0.0387i)3-s + (0.952 + 0.305i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.323 − 0.946i)7-s + (0.893 + 0.448i)8-s + (0.996 − 0.0774i)9-s + (−0.790 + 0.612i)10-s + (0.533 + 0.845i)11-s + (−0.963 − 0.268i)12-s + (−0.0581 + 0.998i)13-s + (0.466 − 0.884i)14-s + (0.657 − 0.753i)15-s + (0.813 + 0.581i)16-s + (−0.835 + 0.549i)17-s + ⋯
L(s,χ)  = 1  + (0.987 + 0.154i)2-s + (−0.999 + 0.0387i)3-s + (0.952 + 0.305i)4-s + (−0.686 + 0.727i)5-s + (−0.993 − 0.116i)6-s + (0.323 − 0.946i)7-s + (0.893 + 0.448i)8-s + (0.996 − 0.0774i)9-s + (−0.790 + 0.612i)10-s + (0.533 + 0.845i)11-s + (−0.963 − 0.268i)12-s + (−0.0581 + 0.998i)13-s + (0.466 − 0.884i)14-s + (0.657 − 0.753i)15-s + (0.813 + 0.581i)16-s + (−0.835 + 0.549i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.557 + 0.829i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.557 + 0.829i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(163\)
\( \varepsilon \)  =  $0.557 + 0.829i$
motivic weight  =  \(0\)
character  :  $\chi_{163} (16, \cdot )$
Sato-Tate  :  $\mu(81)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 163,\ (0:\ ),\ 0.557 + 0.829i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.274321444 + 0.6788779676i$
$L(\frac12,\chi)$  $\approx$  $1.274321444 + 0.6788779676i$
$L(\chi,1)$  $\approx$  1.299154343 + 0.3725099423i
$L(1,\chi)$  $\approx$  1.299154343 + 0.3725099423i

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

−27.860470255736932791086326806967, −26.991551778680037477622812740813, −24.962610274472407431520094995327, −24.45904530801088787867322911101, −23.764584419692900873548748768028, −22.47504899260587975924947491768, −22.1372704561530842475410613424, −20.93569904345435442453546568230, −19.97201689323488149397222781149, −18.825601226292075432583329758790, −17.61178632907934959905159742283, −16.22301119065797853099768232661, −15.83532222896976252966201716073, −14.6755562407783346063133963555, −13.147961161028358579050991715951, −12.39846511214604189589331718644, −11.53075971523301414676745333927, −10.892046710925464117457464855210, −9.118934240126798393502555253400, −7.6594390425578212145198926474, −6.30749426932811641949448369380, −5.28686871101333295686268008323, −4.56966603619368974750008066644, −3.03513691659587047609953822132, −1.14089407087697564789360974513, 1.81329008450055442311161561063, 3.93104772260117023042155009103, 4.330621867517097639348893158471, 5.89214132652760597047738674499, 7.053273043550237409390665093018, 7.491603063030492695743965553993, 9.89099652734367321451585637249, 11.18454048437495386842089448589, 11.508723033245893328592284482305, 12.69557862049844884626939092818, 13.92331791439112474516078651807, 14.91055187238281751001543230708, 15.86572296605922199249938935364, 16.87263408894037734952308543083, 17.728457375092402675407253125510, 19.19673830876402495424932560045, 20.2200072359823015704372078914, 21.37194793898600422780676059051, 22.41511242934454215858776202042, 22.95131483325106957998630464986, 23.77701957765746962729310177232, 24.475375996754580870846885271735, 26.057817577068427962201804997956, 26.83375546679032611199858104264, 27.99055355331876066223119031116

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