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Properties

Label 1-229-229.20-r0-0-0
Degree $1$
Conductor $229$
Sign $-0.806 - 0.590i$
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  + (0.789 − 0.614i)2-s + (−0.191 − 0.981i)3-s + (0.245 − 0.969i)4-s + (0.451 + 0.892i)5-s + (−0.754 − 0.656i)6-s + (−0.298 − 0.954i)7-s + (−0.401 − 0.915i)8-s + (−0.926 + 0.376i)9-s + (0.904 + 0.426i)10-s + (−0.986 + 0.164i)11-s + (−0.998 − 0.0550i)12-s + (0.546 − 0.837i)13-s + (−0.821 − 0.569i)14-s + (0.789 − 0.614i)15-s + (−0.879 − 0.475i)16-s + (0.546 − 0.837i)17-s + ⋯
L(s)  = 1  + (0.789 − 0.614i)2-s + (−0.191 − 0.981i)3-s + (0.245 − 0.969i)4-s + (0.451 + 0.892i)5-s + (−0.754 − 0.656i)6-s + (−0.298 − 0.954i)7-s + (−0.401 − 0.915i)8-s + (−0.926 + 0.376i)9-s + (0.904 + 0.426i)10-s + (−0.986 + 0.164i)11-s + (−0.998 − 0.0550i)12-s + (0.546 − 0.837i)13-s + (−0.821 − 0.569i)14-s + (0.789 − 0.614i)15-s + (−0.879 − 0.475i)16-s + (0.546 − 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.806 - 0.590i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ -0.806 - 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4917032552 - 1.504385926i\)
\(L(\frac12)\) \(\approx\) \(0.4917032552 - 1.504385926i\)
\(L(1)\) \(\approx\) \(1.012757334 - 0.9952348720i\)
\(L(1)\) \(\approx\) \(1.012757334 - 0.9952348720i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.789 - 0.614i)T \)
3 \( 1 + (-0.191 - 0.981i)T \)
5 \( 1 + (0.451 + 0.892i)T \)
7 \( 1 + (-0.298 - 0.954i)T \)
11 \( 1 + (-0.986 + 0.164i)T \)
13 \( 1 + (0.546 - 0.837i)T \)
17 \( 1 + (0.546 - 0.837i)T \)
19 \( 1 + (-0.998 + 0.0550i)T \)
23 \( 1 + (0.904 - 0.426i)T \)
29 \( 1 + (0.975 + 0.218i)T \)
31 \( 1 + (0.350 - 0.936i)T \)
37 \( 1 + (0.851 + 0.523i)T \)
41 \( 1 + (0.137 + 0.990i)T \)
43 \( 1 + (-0.879 + 0.475i)T \)
47 \( 1 + (-0.926 + 0.376i)T \)
53 \( 1 + (0.945 + 0.324i)T \)
59 \( 1 + (0.851 - 0.523i)T \)
61 \( 1 + (0.789 + 0.614i)T \)
67 \( 1 + (0.137 - 0.990i)T \)
71 \( 1 + (0.635 - 0.771i)T \)
73 \( 1 + (-0.592 + 0.805i)T \)
79 \( 1 + (0.975 - 0.218i)T \)
83 \( 1 + (0.0275 + 0.999i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.993 - 0.110i)T \)
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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

−26.454580658044814645687548968018, −25.6412265750621292492730248091, −25.04747878908794911216956322821, −23.6921213825966056149974110568, −23.23939392680596032884019686426, −21.81675950357179735247434129119, −21.28719277985100160626162749130, −20.93055110212379520472782223536, −19.41384103882840711816026169744, −17.89592855312054656594682995106, −16.9094022542857264972634336046, −16.16122046374550100502714003847, −15.500888406540266135163888184611, −14.54712110728427646566028816695, −13.34400342592201012420385779181, −12.50850231901086109124506784632, −11.51053623849469406294869666086, −10.19824692975638841156207697770, −8.83124709112643634709983608522, −8.41368182118139624123457134872, −6.43524520851855708961967794527, −5.56500386279032148151751628803, −4.86098613109826495934486403958, −3.68899787321745965427474699911, −2.36536438192801799078783821533, 0.92814608856522816550074743572, 2.481421392585852842034761293590, 3.21043601863669138391626510250, 4.913165340822220380219920559900, 6.116905560665274279500811551138, 6.84948284064067100322469639122, 7.95929171183474755874173769550, 9.92656769986137119710358769259, 10.67063607433580323140030698952, 11.46733983220531977895698944115, 12.95636845712786529036506688524, 13.24714446224856339541007026366, 14.22686506066902525480126689505, 15.16327437455178308468386272482, 16.591407515202161237832861329470, 17.91559509119504014396589772318, 18.55792067428106196741741881617, 19.452376328187571923567066920, 20.45187857947144260125915908120, 21.30765944961894735176013137411, 22.726553319019946983239952874189, 23.02374160769568084940245981854, 23.736786546566880926138874646532, 25.06581289769025232571478003740, 25.68324282432590392796848916525

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