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Properties

Label 1-229-229.10-r1-0-0
Degree $1$
Conductor $229$
Sign $0.687 - 0.726i$
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  + (0.969 − 0.245i)2-s + (−0.926 − 0.376i)3-s + (0.879 − 0.475i)4-s + (0.592 + 0.805i)5-s + (−0.990 − 0.137i)6-s + (0.569 − 0.821i)7-s + (0.735 − 0.677i)8-s + (0.716 + 0.697i)9-s + (0.771 + 0.635i)10-s + (−0.945 − 0.324i)11-s + (−0.993 + 0.110i)12-s + (0.915 + 0.401i)13-s + (0.350 − 0.936i)14-s + (−0.245 − 0.969i)15-s + (0.546 − 0.837i)16-s + (−0.401 + 0.915i)17-s + ⋯
L(s)  = 1  + (0.969 − 0.245i)2-s + (−0.926 − 0.376i)3-s + (0.879 − 0.475i)4-s + (0.592 + 0.805i)5-s + (−0.990 − 0.137i)6-s + (0.569 − 0.821i)7-s + (0.735 − 0.677i)8-s + (0.716 + 0.697i)9-s + (0.771 + 0.635i)10-s + (−0.945 − 0.324i)11-s + (−0.993 + 0.110i)12-s + (0.915 + 0.401i)13-s + (0.350 − 0.936i)14-s + (−0.245 − 0.969i)15-s + (0.546 − 0.837i)16-s + (−0.401 + 0.915i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.687 - 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.038479545 - 1.307671521i\)
\(L(\frac12)\) \(\approx\) \(3.038479545 - 1.307671521i\)
\(L(1)\) \(\approx\) \(1.777811898 - 0.4673138267i\)
\(L(1)\) \(\approx\) \(1.777811898 - 0.4673138267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.969 - 0.245i)T \)
3 \( 1 + (-0.926 - 0.376i)T \)
5 \( 1 + (0.592 + 0.805i)T \)
7 \( 1 + (0.569 - 0.821i)T \)
11 \( 1 + (-0.945 - 0.324i)T \)
13 \( 1 + (0.915 + 0.401i)T \)
17 \( 1 + (-0.401 + 0.915i)T \)
19 \( 1 + (0.993 + 0.110i)T \)
23 \( 1 + (0.771 - 0.635i)T \)
29 \( 1 + (-0.426 - 0.904i)T \)
31 \( 1 + (0.656 + 0.754i)T \)
37 \( 1 + (0.451 - 0.892i)T \)
41 \( 1 + (0.272 - 0.962i)T \)
43 \( 1 + (0.546 + 0.837i)T \)
47 \( 1 + (-0.697 + 0.716i)T \)
53 \( 1 + (0.789 - 0.614i)T \)
59 \( 1 + (0.892 - 0.451i)T \)
61 \( 1 + (0.245 - 0.969i)T \)
67 \( 1 + (-0.272 - 0.962i)T \)
71 \( 1 + (0.191 - 0.981i)T \)
73 \( 1 + (0.954 + 0.298i)T \)
79 \( 1 + (-0.426 + 0.904i)T \)
83 \( 1 + (-0.998 - 0.0550i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (-0.975 - 0.218i)T \)
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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

−25.99506438912095304251652941468, −25.022430913587793887374540130903, −24.2785414850387474755406517098, −23.45911780368248060985449346824, −22.53002521089948214098781438288, −21.67491611500988014553005015914, −20.865485976959462597208582905709, −20.421034873341520130740406297930, −18.30795928009335403210548612520, −17.70302170400982579521488463982, −16.535586660189357188697332148016, −15.780036880176358902619370868328, −15.09909423057155434636302262340, −13.563922872384119271036029386701, −12.90197889723213988931929028257, −11.83614780044867694411950703110, −11.142875837253800824242346240716, −9.832168151817509738140451674128, −8.53279917120731680681853042829, −7.17303467281595770066918776710, −5.72461083666042905749848441199, −5.34427413275179456951507346556, −4.47233120524272898371975808466, −2.80077217751379178021106437983, −1.26901168536585948982323800526, 1.06455729134575062891120497453, 2.26243880560063369347264103064, 3.76261931789898997725694438474, 5.008129147192369840678516966593, 5.98513759347763861568019368112, 6.80455315472374028654867495510, 7.821800254436884627433385907941, 10.047442856802526518467342262679, 10.96536199269698266544020207835, 11.21267060763653318263605370227, 12.74835936738751745528977045931, 13.52537497141161795136343754679, 14.22987847908051791902275371480, 15.51567139346716261911677060526, 16.48784303792390443984972060501, 17.59739350224243712983712311488, 18.464663816833999055227330943394, 19.420019665204002823731666589227, 20.98703921053359955380289077299, 21.29626052684864396004366265582, 22.573143485243949166360319342880, 23.06451602063570006025893886207, 23.96870289813785956740674553742, 24.65363063486803875589184402112, 25.980226116339061920694892115325

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