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Properties

Label 1-229-229.5-r0-0-0
Degree $1$
Conductor $229$
Sign $0.453 + 0.891i$
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 + (−0.821 + 0.569i)3-s + (0.945 + 0.324i)4-s + (0.716 + 0.697i)5-s + (−0.904 + 0.426i)6-s + (0.998 − 0.0550i)7-s + (0.879 + 0.475i)8-s + (0.350 − 0.936i)9-s + (0.592 + 0.805i)10-s + (−0.677 − 0.735i)11-s + (−0.962 + 0.272i)12-s + (−0.245 + 0.969i)13-s + (0.993 + 0.110i)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 + (−0.821 + 0.569i)3-s + (0.945 + 0.324i)4-s + (0.716 + 0.697i)5-s + (−0.904 + 0.426i)6-s + (0.998 − 0.0550i)7-s + (0.879 + 0.475i)8-s + (0.350 − 0.936i)9-s + (0.592 + 0.805i)10-s + (−0.677 − 0.735i)11-s + (−0.962 + 0.272i)12-s + (−0.245 + 0.969i)13-s + (0.993 + 0.110i)14-s + (−0.986 − 0.164i)15-s + (0.789 + 0.614i)16-s + (0.245 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.453 + 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.722744963 + 1.056895990i\)
\(L(\frac12)\) \(\approx\) \(1.722744963 + 1.056895990i\)
\(L(1)\) \(\approx\) \(1.597857276 + 0.6114539817i\)
\(L(1)\) \(\approx\) \(1.597857276 + 0.6114539817i\)

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 \( 1 + (-0.821 + 0.569i)T \)
5 \( 1 + (0.716 + 0.697i)T \)
7 \( 1 + (0.998 - 0.0550i)T \)
11 \( 1 + (-0.677 - 0.735i)T \)
13 \( 1 + (-0.245 + 0.969i)T \)
17 \( 1 + (0.245 - 0.969i)T \)
19 \( 1 + (-0.962 - 0.272i)T \)
23 \( 1 + (0.592 - 0.805i)T \)
29 \( 1 + (-0.451 + 0.892i)T \)
31 \( 1 + (-0.975 + 0.218i)T \)
37 \( 1 + (-0.926 - 0.376i)T \)
41 \( 1 + (-0.635 + 0.771i)T \)
43 \( 1 + (0.789 - 0.614i)T \)
47 \( 1 + (-0.350 + 0.936i)T \)
53 \( 1 + (-0.0825 - 0.996i)T \)
59 \( 1 + (0.926 - 0.376i)T \)
61 \( 1 + (-0.986 + 0.164i)T \)
67 \( 1 + (-0.635 - 0.771i)T \)
71 \( 1 + (-0.298 - 0.954i)T \)
73 \( 1 + (-0.0275 - 0.999i)T \)
79 \( 1 + (-0.451 - 0.892i)T \)
83 \( 1 + (0.137 - 0.990i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.851 + 0.523i)T \)
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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

−25.67990857043321566911226646118, −24.98318582846429643612283272782, −24.11912686358399768922848776933, −23.53524427266832896638338089160, −22.59296054120140226742226452897, −21.50084453857509960265992701491, −20.94920212899482960503386610668, −19.92551842844943801326673114119, −18.64734099408186904748540180101, −17.40554333457073290832405062492, −17.05242445472323338623355634775, −15.59202985346815419828431905813, −14.6999186793149073185858103504, −13.39112329221537113761572549082, −12.81226305969157319953621559951, −12.05332718533154296806516861603, −10.84628984209558756793190491882, −10.14225334094878563199140228116, −8.19348454345969836731930092478, −7.22213848724242165385567412956, −5.75622566225965611470177061247, −5.33819133287996965660491114077, −4.30537877277165420505536310880, −2.24990553645528470365436091578, −1.45053808903926307120740330486, 1.91109856822752573853526007357, 3.25815797994995815837396343296, 4.67448686990877452830429741482, 5.34154625262137632899812487898, 6.42381878949425925210906038722, 7.319845941076576910006108954661, 9.014017081429960650600564606799, 10.63372363067810041332198532296, 10.9730495970523842010545680490, 11.98664440031005430769277753470, 13.23616782084186671500131448761, 14.34124822342407547701723342949, 14.85930826735046703524473435986, 16.134829422762423072438740132708, 16.87330337377777587722732007, 17.8775286864433798152374051248, 18.87174282443452452836324025203, 20.68751657526500560850919278692, 21.222071456375753880136995792774, 21.85738223578339003663308329009, 22.718006262441022745124192412972, 23.7271615934226045696098176072, 24.283715330219135100339895470244, 25.57198389165168330725987596689, 26.49815067129909671337785483867

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