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Properties

Label 1-229-229.99-r0-0-0
Degree $1$
Conductor $229$
Sign $0.343 - 0.939i$
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  + (0.0825 + 0.996i)2-s + (−0.298 + 0.954i)3-s + (−0.986 + 0.164i)4-s + (−0.926 + 0.376i)5-s + (−0.975 − 0.218i)6-s + (−0.0275 + 0.999i)7-s + (−0.245 − 0.969i)8-s + (−0.821 − 0.569i)9-s + (−0.451 − 0.892i)10-s + (−0.401 − 0.915i)11-s + (0.137 − 0.990i)12-s + (−0.789 − 0.614i)13-s + (−0.998 + 0.0550i)14-s + (−0.0825 − 0.996i)15-s + (0.945 − 0.324i)16-s + (0.789 + 0.614i)17-s + ⋯
L(s)  = 1  + (0.0825 + 0.996i)2-s + (−0.298 + 0.954i)3-s + (−0.986 + 0.164i)4-s + (−0.926 + 0.376i)5-s + (−0.975 − 0.218i)6-s + (−0.0275 + 0.999i)7-s + (−0.245 − 0.969i)8-s + (−0.821 − 0.569i)9-s + (−0.451 − 0.892i)10-s + (−0.401 − 0.915i)11-s + (0.137 − 0.990i)12-s + (−0.789 − 0.614i)13-s + (−0.998 + 0.0550i)14-s + (−0.0825 − 0.996i)15-s + (0.945 − 0.324i)16-s + (0.789 + 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $0.343 - 0.939i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ 0.343 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1585710804 + 0.1108951174i\)
\(L(\frac12)\) \(\approx\) \(-0.1585710804 + 0.1108951174i\)
\(L(1)\) \(\approx\) \(0.3191061712 + 0.4562580261i\)
\(L(1)\) \(\approx\) \(0.3191061712 + 0.4562580261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (0.0825 + 0.996i)T \)
3 \( 1 + (-0.298 + 0.954i)T \)
5 \( 1 + (-0.926 + 0.376i)T \)
7 \( 1 + (-0.0275 + 0.999i)T \)
11 \( 1 + (-0.401 - 0.915i)T \)
13 \( 1 + (-0.789 - 0.614i)T \)
17 \( 1 + (0.789 + 0.614i)T \)
19 \( 1 + (0.137 + 0.990i)T \)
23 \( 1 + (-0.451 + 0.892i)T \)
29 \( 1 + (-0.851 - 0.523i)T \)
31 \( 1 + (-0.993 - 0.110i)T \)
37 \( 1 + (-0.191 - 0.981i)T \)
41 \( 1 + (-0.904 - 0.426i)T \)
43 \( 1 + (0.945 + 0.324i)T \)
47 \( 1 + (0.821 + 0.569i)T \)
53 \( 1 + (-0.677 - 0.735i)T \)
59 \( 1 + (0.191 - 0.981i)T \)
61 \( 1 + (-0.0825 + 0.996i)T \)
67 \( 1 + (-0.904 + 0.426i)T \)
71 \( 1 + (-0.592 - 0.805i)T \)
73 \( 1 + (-0.716 + 0.697i)T \)
79 \( 1 + (-0.851 + 0.523i)T \)
83 \( 1 + (-0.754 - 0.656i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.962 + 0.272i)T \)
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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

−25.78165156178453429922546163753, −24.15929306563227061770194493771, −23.69467070397085441392778026417, −22.92879126356234665250981675548, −22.090924901091301476206119427432, −20.33769141879150176567754026915, −20.23576937319575089681512581396, −19.14071777959279270310197443412, −18.3830915646613014775576445923, −17.280482461976123053419020726826, −16.50450717335874663548383227197, −14.80345970965682539191456226103, −13.82209366812368111494176466796, −12.82343816428390812122647174415, −12.15370410615828943725571522821, −11.32719449311213721095305777508, −10.29409193320634994504085103630, −9.013670267176525573792208224742, −7.689942103757726524683679588259, −7.08533248620896084164509204506, −5.145573606677028631010675647297, −4.32999063206225483783339728999, −2.89499121981292789294913305402, −1.51452941847006246754573422911, −0.14657135949673299325271822867, 3.15345876404886861547104111436, 3.96896023771795933138485676767, 5.46608016603457821033359984763, 5.840783771455371831828559924700, 7.57061532546696973815092572281, 8.34860269474934279858208945925, 9.4481771660653686932454997248, 10.52830104682042899080602843875, 11.77353549054920503403481050856, 12.66212049702938984099186471008, 14.36585936542967105004297655697, 14.94972169886724430617436735357, 15.78878221724950683649878583507, 16.37345079749826440369243764572, 17.47477493664936990313922995392, 18.601526651820909377406535535108, 19.36008516183150683852403092590, 20.910395894308877370659047595186, 21.943945589282120069859860061174, 22.42874953965210112468000510203, 23.43370438400394978032051888768, 24.22588989326575588850620913398, 25.402611126198445172327395973568, 26.22280662648112363114643623661, 27.233006219477257898247112445433

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