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Properties

Label 1-229-229.12-r0-0-0
Degree $1$
Conductor $229$
Sign $0.312 + 0.949i$
Analytic cond. $1.06347$
Root an. cond. $1.06347$
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  + (0.986 + 0.164i)2-s + (0.904 + 0.426i)3-s + (0.945 + 0.324i)4-s + (−0.962 + 0.272i)5-s + (0.821 + 0.569i)6-s + (−0.451 + 0.892i)7-s + (0.879 + 0.475i)8-s + (0.635 + 0.771i)9-s + (−0.993 + 0.110i)10-s + (−0.677 − 0.735i)11-s + (0.716 + 0.697i)12-s + (−0.245 + 0.969i)13-s + (−0.592 + 0.805i)14-s + (−0.986 − 0.164i)15-s + (0.789 + 0.614i)16-s + (0.245 − 0.969i)17-s + ⋯
L(s)  = 1  + (0.986 + 0.164i)2-s + (0.904 + 0.426i)3-s + (0.945 + 0.324i)4-s + (−0.962 + 0.272i)5-s + (0.821 + 0.569i)6-s + (−0.451 + 0.892i)7-s + (0.879 + 0.475i)8-s + (0.635 + 0.771i)9-s + (−0.993 + 0.110i)10-s + (−0.677 − 0.735i)11-s + (0.716 + 0.697i)12-s + (−0.245 + 0.969i)13-s + (−0.592 + 0.805i)14-s + (−0.986 − 0.164i)15-s + (0.789 + 0.614i)16-s + (0.245 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.312 + 0.949i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.312 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.927193145 + 1.394218070i\)
\(L(\frac12)\) \(\approx\) \(1.927193145 + 1.394218070i\)
\(L(1)\) \(\approx\) \(1.846434868 + 0.7678851665i\)
\(L(1)\) \(\approx\) \(1.846434868 + 0.7678851665i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.986 + 0.164i)T \)
3 \( 1 + (0.904 + 0.426i)T \)
5 \( 1 + (-0.962 + 0.272i)T \)
7 \( 1 + (-0.451 + 0.892i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (-0.245 + 0.969i)T \)
17 \( 1 + (0.245 - 0.969i)T \)
19 \( 1 + (0.716 - 0.697i)T \)
23 \( 1 + (-0.993 - 0.110i)T \)
29 \( 1 + (0.998 - 0.0550i)T \)
31 \( 1 + (0.298 - 0.954i)T \)
37 \( 1 + (0.137 + 0.990i)T \)
41 \( 1 + (-0.350 - 0.936i)T \)
43 \( 1 + (0.789 - 0.614i)T \)
47 \( 1 + (-0.635 - 0.771i)T \)
53 \( 1 + (-0.0825 - 0.996i)T \)
59 \( 1 + (-0.137 + 0.990i)T \)
61 \( 1 + (-0.986 + 0.164i)T \)
67 \( 1 + (-0.350 + 0.936i)T \)
71 \( 1 + (0.975 + 0.218i)T \)
73 \( 1 + (-0.851 + 0.523i)T \)
79 \( 1 + (0.998 + 0.0550i)T \)
83 \( 1 + (-0.926 + 0.376i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.0275 - 0.999i)T \)
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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

−26.00958746182898940493003515302, −25.094830538041218892124698108834, −24.16919746120314013642496963122, −23.32879462213862995782693824982, −22.85838926465534310752263537384, −21.37342442075194111741601976921, −20.31114952060280148637918340580, −19.95920636504635974601827049892, −19.21170454067908304310685886350, −17.81644042943132109418033411110, −16.24794693182427704034851573301, −15.54479067951478754529417878800, −14.63509290948411159346314020094, −13.7213110834349474621054433657, −12.56641582864193923966590061849, −12.39146435089363368099002542709, −10.68409334646613091961124520137, −9.84822332172256684716095921157, −7.90010160264999523777956902282, −7.6284811394237176889447614492, −6.328956032476743759974421459350, −4.72242682110039263059003364483, −3.72418792072433662140818930374, −2.93023476522834929921331995073, −1.3181090395075095199911645854, 2.46395957141073415522901221636, 3.11523640038534284688603700537, 4.23206620057339066524775106893, 5.28838115300180036929799851504, 6.75430024781223814307657036515, 7.77488741494718462817849407580, 8.731005846049895698918349945373, 10.07128292422187852771061436812, 11.43673221132064070128348879774, 12.099962659772160149337691856259, 13.4388747091679683991582777012, 14.14719721551765730582029922528, 15.22122078792629432719892575941, 15.87462198517186019216720703275, 16.36832656125654513102050852953, 18.51687867756815406030322807410, 19.26605591496720632108014262668, 20.142391691802528430082048223998, 21.12156026087697030039081051257, 21.95671530326877372519469600928, 22.63962009376411524156764045131, 23.992822980838117950569623428093, 24.44592871043967755135229804967, 25.704385082085032548900283348792, 26.27881432049886400846313296870

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