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Properties

Label 1-229-229.141-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.136 + 0.990i$
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  + (0.164 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.136 + 0.990i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.136 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293246897 + 1.483913952i\)
\(L(\frac12)\) \(\approx\) \(1.293246897 + 1.483913952i\)
\(L(1)\) \(\approx\) \(0.9493627062 + 0.7185450561i\)
\(L(1)\) \(\approx\) \(0.9493627062 + 0.7185450561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.164 + 0.986i)T \)
3 \( 1 + (-0.0825 + 0.996i)T \)
5 \( 1 + (-0.245 - 0.969i)T \)
7 \( 1 + (0.837 + 0.546i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (0.969 - 0.245i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (0.245 - 0.969i)T \)
23 \( 1 + (0.915 + 0.401i)T \)
29 \( 1 + (-0.837 - 0.546i)T \)
31 \( 1 + (0.735 + 0.677i)T \)
37 \( 1 + (0.789 + 0.614i)T \)
41 \( 1 + (-0.164 - 0.986i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (0.164 - 0.986i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (-0.614 - 0.789i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (0.164 - 0.986i)T \)
71 \( 1 + (0.677 + 0.735i)T \)
73 \( 1 + (0.475 + 0.879i)T \)
79 \( 1 + (-0.837 + 0.546i)T \)
83 \( 1 + (0.789 - 0.614i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.879 + 0.475i)T \)
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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

−25.97138194776118882272567695984, −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