Properties

Degree 1
Conductor 229
Sign $0.822 + 0.569i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.822 + 0.569i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.822 + 0.569i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(229\)
\( \varepsilon \)  =  $0.822 + 0.569i$
motivic weight  =  \(0\)
character  :  $\chi_{229} (84, \cdot )$
Sato-Tate  :  $\mu(76)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 229,\ (1:\ ),\ 0.822 + 0.569i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2587151322 + 0.08085540843i$
$L(\frac12,\chi)$  $\approx$  $0.2587151322 + 0.08085540843i$
$L(\chi,1)$  $\approx$  0.6504551656 - 0.5287950232i
$L(1,\chi)$  $\approx$  0.6504551656 - 0.5287950232i

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

−25.90903944330878721654507749541, −25.28904564895792982086671808955, −23.85041786705217706896043548439, −23.11609106730454974635140465020, −22.612587972496302034273669975731, −21.66464283874422037236206480505, −20.55019104001432144482621074535, −19.91710610251678948876772142791, −18.28921348328722757255545964253, −17.15099731993252313016337342947, −16.23147082083761049622038323615, −15.49658443318177243026750397265, −15.188521699296088700367775022898, −13.643774522176376812227564948139, −12.412117453103459529304271003807, −11.96291469701072721267348232021, −10.54018239176151935265271369719, −9.390759084837598603598024635664, −8.166145179709049194733443218263, −7.14615812150090517262017389634, −5.742943882656059535863988146941, −5.04237832740185225313158007636, −3.75575906026728135425500100698, −3.133378844077320813719264667521, −0.08337063808068159701488847386, 1.112759904766906114522779176125, 2.82492738946957058632792988382, 3.62145621736649849815632437285, 5.1444507261276316772702963643, 6.32747041655134664130891941369, 7.1016635895467828255377099910, 8.44967630212849386119572825493, 10.06451178020846962983604922989, 11.06104646485385255924143342165, 11.92487073810885118146516367431, 12.58318559650226396673100067893, 13.71928818965484796292849015927, 14.32520658168176576011225685070, 15.989867663659331734508073265532, 16.397054518073287951847484484685, 18.40386520363064224926843464641, 18.80805340446045574533337327684, 19.574191310048537866103650088354, 20.48156820280738700765465064125, 21.85453424898510251864109492671, 22.65604012298104063196744283941, 23.46658097100479541332409612133, 23.86270509620918648189984728412, 25.00423960999377548717670177901, 26.26241272201195594823378956483

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