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Properties

Label 1-229-229.147-r0-0-0
Degree $1$
Conductor $229$
Sign $0.366 - 0.930i$
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  + (0.677 − 0.735i)2-s + (0.993 − 0.110i)3-s + (−0.0825 − 0.996i)4-s + (−0.754 + 0.656i)5-s + (0.592 − 0.805i)6-s + (0.962 + 0.272i)7-s + (−0.789 − 0.614i)8-s + (0.975 − 0.218i)9-s + (−0.0275 + 0.999i)10-s + (0.546 − 0.837i)11-s + (−0.191 − 0.981i)12-s + (−0.945 − 0.324i)13-s + (0.851 − 0.523i)14-s + (−0.677 + 0.735i)15-s + (−0.986 + 0.164i)16-s + (0.945 + 0.324i)17-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)2-s + (0.993 − 0.110i)3-s + (−0.0825 − 0.996i)4-s + (−0.754 + 0.656i)5-s + (0.592 − 0.805i)6-s + (0.962 + 0.272i)7-s + (−0.789 − 0.614i)8-s + (0.975 − 0.218i)9-s + (−0.0275 + 0.999i)10-s + (0.546 − 0.837i)11-s + (−0.191 − 0.981i)12-s + (−0.945 − 0.324i)13-s + (0.851 − 0.523i)14-s + (−0.677 + 0.735i)15-s + (−0.986 + 0.164i)16-s + (0.945 + 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.366 - 0.930i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.366 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.813741492 - 1.235411265i\)
\(L(\frac12)\) \(\approx\) \(1.813741492 - 1.235411265i\)
\(L(1)\) \(\approx\) \(1.673859998 - 0.7709797294i\)
\(L(1)\) \(\approx\) \(1.673859998 - 0.7709797294i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.677 - 0.735i)T \)
3 \( 1 + (0.993 - 0.110i)T \)
5 \( 1 + (-0.754 + 0.656i)T \)
7 \( 1 + (0.962 + 0.272i)T \)
11 \( 1 + (0.546 - 0.837i)T \)
13 \( 1 + (-0.945 - 0.324i)T \)
17 \( 1 + (0.945 + 0.324i)T \)
19 \( 1 + (-0.191 + 0.981i)T \)
23 \( 1 + (-0.0275 - 0.999i)T \)
29 \( 1 + (-0.716 + 0.697i)T \)
31 \( 1 + (-0.451 - 0.892i)T \)
37 \( 1 + (0.350 + 0.936i)T \)
41 \( 1 + (0.298 + 0.954i)T \)
43 \( 1 + (-0.986 - 0.164i)T \)
47 \( 1 + (-0.975 + 0.218i)T \)
53 \( 1 + (-0.401 + 0.915i)T \)
59 \( 1 + (-0.350 + 0.936i)T \)
61 \( 1 + (-0.677 - 0.735i)T \)
67 \( 1 + (0.298 - 0.954i)T \)
71 \( 1 + (-0.998 + 0.0550i)T \)
73 \( 1 + (-0.137 + 0.990i)T \)
79 \( 1 + (-0.716 - 0.697i)T \)
83 \( 1 + (0.635 + 0.771i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.926 - 0.376i)T \)
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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

−26.46372392701246296374738227226, −25.38034465021292490161592075958, −24.63448260998294796452814327429, −23.95231254525970426813778820743, −23.13099074849952136812544680384, −21.79030755301726229709449043912, −20.99147228861928658680601190479, −20.17302879673177948653710132430, −19.35866286279772548853787167417, −17.81375182073564479023559922548, −16.91610832234697603764510258466, −15.8761799723403454153647847458, −14.90626774845800054674593987865, −14.46205802193202328766369584548, −13.32595019496670931501191231470, −12.32821671151429756379417319072, −11.47412561278674971419528715347, −9.58246244304008683434537278923, −8.67633313662048187898239266335, −7.54875208648562339170031821860, −7.24642874909202116784552800571, −5.10734856163261120089719742623, −4.45455425389295613754777699689, −3.47257678125419917966386854336, −1.888971719294578718633985880426, 1.493523533297560499439771433464, 2.76994449601542776365735384390, 3.6291676305055857076092371839, 4.6736078370150726543616091250, 6.16636532816246537078505832793, 7.59694458304566035149077609538, 8.45882130393406082616751310340, 9.80454361692850611549389650880, 10.813868876733492018457355944299, 11.83915846296251158415116776124, 12.635239068110533225395429010466, 14.00531634777397728462288404002, 14.77977931681025730229260684146, 14.952169772699433866616986919771, 16.55582590859937968747936823347, 18.44928140779970084995897554645, 18.78745846447046866878963908290, 19.79214326992452641476995991700, 20.52953399613817342563045134908, 21.53352001706842671397470843509, 22.2318464199883174591693195439, 23.451994851925786403537302541213, 24.30750230210696808688531374909, 24.93780870068543873693807943837, 26.39243651126670686239830863023

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