Properties

Degree 1
Conductor 229
Sign $-0.136 + 0.990i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.164 + 0.986i)2-s + (−0.0825 + 0.996i)3-s + (−0.945 + 0.324i)4-s + (−0.245 − 0.969i)5-s + (−0.996 + 0.0825i)6-s + (0.837 + 0.546i)7-s + (−0.475 − 0.879i)8-s + (−0.986 − 0.164i)9-s + (0.915 − 0.401i)10-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)12-s + (0.969 − 0.245i)13-s + (−0.401 + 0.915i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s,χ)  = 1  + (0.164 + 0.986i)2-s + (−0.0825 + 0.996i)3-s + (−0.945 + 0.324i)4-s + (−0.245 − 0.969i)5-s + (−0.996 + 0.0825i)6-s + (0.837 + 0.546i)7-s + (−0.475 − 0.879i)8-s + (−0.986 − 0.164i)9-s + (0.915 − 0.401i)10-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)12-s + (0.969 − 0.245i)13-s + (−0.401 + 0.915i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.136 + 0.990i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.136 + 0.990i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $-0.136 + 0.990i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (141, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ -0.136 + 0.990i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.293246897 + 1.483913952i$
$L(\frac12,\chi)$  $\approx$  $1.293246897 + 1.483913952i$
$L(\chi,1)$  $\approx$  0.9493627062 + 0.7185450561i
$L(1,\chi)$  $\approx$  0.9493627062 + 0.7185450561i

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

−25.97138194776118882272567695984, −24.86852118956906860278043583845, −23.688543218011961856337467464428, −22.93635504381700963691281798504, −22.50276831244241677641387802243, −20.96290905196591756969828319887, −20.298236399779471042797488550431, −19.24816548710262768917388271256, −18.41074340062594177932016284716, −17.93419064078690744175831879255, −16.81212793151740603212053244333, −14.82507650343000857631428603443, −14.2324451208359921526289367023, −13.42368300536597172055192879561, −12.189222076971773925497702255391, −11.395063240488527582523526224076, −10.77642893143905088121030158759, −9.40080805050254274654678841949, −8.06106719383453031514088551743, −7.140230862757867072231516083885, −5.910358862737037791390471885436, −4.41414583748365410008283214538, −3.22266446032148225321512768721, −1.96076254052787042507365016267, −0.93960960819565236949373588348, 0.93468599325774281554379654695, 3.444623567817908598217440898681, 4.42892411468280962125973449420, 5.33598311322290533985006687633, 6.132226593953783785887261482595, 7.97613616414740231333297666078, 8.73838698865783095832414672271, 9.28833983267194597646854829801, 10.95446719080804534993937274756, 11.91504261919000663861957947094, 13.21926864612162921586895772059, 14.22856935393937939733501052148, 15.34027868766040892119151249102, 15.74476719562089457171252632992, 16.98190701276872684944019581335, 17.30131852697801433766659197168, 18.72722900144315514443768797567, 20.0058750815681846185387848323, 21.264279726834756589627985418916, 21.565714384793739674972219914649, 22.805915032489368940717774354762, 23.74248217885552447383835497946, 24.535168953324700022411139292033, 25.37860259124955559596269064836, 26.39707371463820181017226561886

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