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Properties

Label 1-229-229.2-r1-0-0
Degree $1$
Conductor $229$
Sign $0.902 - 0.431i$
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  + (0.915 − 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (0.837 + 0.546i)6-s + (0.614 − 0.789i)7-s + (0.324 − 0.945i)8-s + (−0.401 + 0.915i)9-s + (0.969 − 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (0.164 − 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯
L(s)  = 1  + (0.915 − 0.401i)2-s + (0.546 + 0.837i)3-s + (0.677 − 0.735i)4-s + (0.986 + 0.164i)5-s + (0.837 + 0.546i)6-s + (0.614 − 0.789i)7-s + (0.324 − 0.945i)8-s + (−0.401 + 0.915i)9-s + (0.969 − 0.245i)10-s + (0.879 − 0.475i)11-s + (0.986 + 0.164i)12-s + (0.164 − 0.986i)13-s + (0.245 − 0.969i)14-s + (0.401 + 0.915i)15-s + (−0.0825 − 0.996i)16-s + (−0.986 − 0.164i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.902 - 0.431i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ 0.902 - 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.074357395 - 1.150495049i\)
\(L(\frac12)\) \(\approx\) \(5.074357395 - 1.150495049i\)
\(L(1)\) \(\approx\) \(2.676789902 - 0.3546452485i\)
\(L(1)\) \(\approx\) \(2.676789902 - 0.3546452485i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.915 - 0.401i)T \)
3 \( 1 + (0.546 + 0.837i)T \)
5 \( 1 + (0.986 + 0.164i)T \)
7 \( 1 + (0.614 - 0.789i)T \)
11 \( 1 + (0.879 - 0.475i)T \)
13 \( 1 + (0.164 - 0.986i)T \)
17 \( 1 + (-0.986 - 0.164i)T \)
19 \( 1 + (-0.986 + 0.164i)T \)
23 \( 1 + (0.969 + 0.245i)T \)
29 \( 1 + (-0.614 + 0.789i)T \)
31 \( 1 + (-0.475 - 0.879i)T \)
37 \( 1 + (-0.0825 + 0.996i)T \)
41 \( 1 + (-0.915 + 0.401i)T \)
43 \( 1 + (-0.0825 + 0.996i)T \)
47 \( 1 + (0.915 + 0.401i)T \)
53 \( 1 + (0.546 + 0.837i)T \)
59 \( 1 + (0.996 - 0.0825i)T \)
61 \( 1 + (-0.401 + 0.915i)T \)
67 \( 1 + (0.915 + 0.401i)T \)
71 \( 1 + (0.879 + 0.475i)T \)
73 \( 1 + (-0.324 + 0.945i)T \)
79 \( 1 + (-0.614 - 0.789i)T \)
83 \( 1 + (-0.0825 - 0.996i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.945 + 0.324i)T \)
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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

−25.657964179304417004314399482667, −25.131282404235211021841507930256, −24.44948106356579348284587842341, −23.71436850010342322956131089991, −22.47700863787490695770415278925, −21.5123850810885364999333615361, −20.88958630876927479205288351796, −19.82369893060580896621045988690, −18.63970582835638634816738987699, −17.52564804452986812289633216848, −16.96229857072670679268853250481, −15.31568378612945672757604159876, −14.57622502529307817392854207357, −13.843490053711718492992744935308, −12.91718756675623604866151223066, −12.127957871889269632702298142574, −11.09138470259255868559898679968, −9.08205936198547427256279444194, −8.60848874186309360571163264518, −6.98473381360711839840247295019, −6.42383415194346660407674127460, −5.24498747171499907684393050345, −3.97454578836502336024980995723, −2.258293236779136816409496315652, −1.84540168540636053065061532668, 1.3523403351341731842096097442, 2.628058246062175048092214061535, 3.75186958234696131445310685892, 4.747263896238294016377605983162, 5.778673539014493484503812409754, 7.00741108369895414355735670856, 8.615148101261376508050140787290, 9.74855411809167933454469997409, 10.72281772129513083403118494132, 11.226464480029595729317554393747, 13.07917447072725545483987412460, 13.622274812502157940800384262694, 14.623844823774174396071191017213, 15.11489548833171650784240489580, 16.560631631971499951452406703661, 17.34514479196570864446004193542, 18.852306445337739684049054350736, 20.10188612363684125187152017382, 20.50501131913171611469587958698, 21.51633070523548200868606919929, 22.10209643410071839891832983823, 22.98767513745761828533802377407, 24.30777489687038629571870500051, 25.07265708317535252192319556410, 25.899154182428011748986087126455

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