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Properties

Label 1-229-229.84-r1-0-0
Degree $1$
Conductor $229$
Sign $0.822 + 0.569i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (0.735 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.822 + 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2587151322 + 0.08085540843i\)
\(L(\frac12)\) \(\approx\) \(0.2587151322 + 0.08085540843i\)
\(L(1)\) \(\approx\) \(0.6504551656 - 0.5287950232i\)
\(L(1)\) \(\approx\) \(0.6504551656 - 0.5287950232i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.735 - 0.677i)T \)
3 \( 1 + (-0.401 - 0.915i)T \)
5 \( 1 + (-0.945 + 0.324i)T \)
7 \( 1 + (-0.969 - 0.245i)T \)
11 \( 1 + (-0.546 - 0.837i)T \)
13 \( 1 + (0.324 + 0.945i)T \)
17 \( 1 + (0.945 - 0.324i)T \)
19 \( 1 + (0.945 + 0.324i)T \)
23 \( 1 + (-0.475 - 0.879i)T \)
29 \( 1 + (0.969 + 0.245i)T \)
31 \( 1 + (-0.837 + 0.546i)T \)
37 \( 1 + (-0.986 + 0.164i)T \)
41 \( 1 + (-0.735 + 0.677i)T \)
43 \( 1 + (-0.986 + 0.164i)T \)
47 \( 1 + (0.735 + 0.677i)T \)
53 \( 1 + (-0.401 - 0.915i)T \)
59 \( 1 + (0.164 - 0.986i)T \)
61 \( 1 + (-0.677 + 0.735i)T \)
67 \( 1 + (0.735 + 0.677i)T \)
71 \( 1 + (-0.546 + 0.837i)T \)
73 \( 1 + (0.614 + 0.789i)T \)
79 \( 1 + (0.969 - 0.245i)T \)
83 \( 1 + (-0.986 - 0.164i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.789 - 0.614i)T \)
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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

−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