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Properties

Label 1-229-229.15-r0-0-0
Degree $1$
Conductor $229$
Sign $-0.661 - 0.749i$
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  + (0.401 + 0.915i)2-s + (0.546 + 0.837i)3-s + (−0.677 + 0.735i)4-s + (−0.986 − 0.164i)5-s + (−0.546 + 0.837i)6-s + (−0.789 − 0.614i)7-s + (−0.945 − 0.324i)8-s + (−0.401 + 0.915i)9-s + (−0.245 − 0.969i)10-s + (−0.879 + 0.475i)11-s + (−0.986 − 0.164i)12-s + (0.986 + 0.164i)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  + (0.401 + 0.915i)2-s + (0.546 + 0.837i)3-s + (−0.677 + 0.735i)4-s + (−0.986 − 0.164i)5-s + (−0.546 + 0.837i)6-s + (−0.789 − 0.614i)7-s + (−0.945 − 0.324i)8-s + (−0.401 + 0.915i)9-s + (−0.245 − 0.969i)10-s + (−0.879 + 0.475i)11-s + (−0.986 − 0.164i)12-s + (0.986 + 0.164i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(1.06347\)
Root analytic conductor: \(1.06347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (0:\ ),\ -0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2488779135 + 0.5518499710i\)
\(L(\frac12)\) \(\approx\) \(-0.2488779135 + 0.5518499710i\)
\(L(1)\) \(\approx\) \(0.5205572475 + 0.6614328871i\)
\(L(1)\) \(\approx\) \(0.5205572475 + 0.6614328871i\)

Euler product

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

−25.95925737331022527660922549446, −24.58721691891277587604936709287, −23.726963658195732179855426325160, −23.071042796023445914901389465403, −22.13027919417758095981278606704, −20.88824341024728743362314127129, −20.10164537006885937880210720782, −19.09093793085624119783676941369, −18.81365290821767451257537395398, −17.84118797743704224759521457330, −15.91110090082840016652171879656, −15.204073377762207347237051549189, −14.062016143173011962410907371685, −12.91660172839976855754231805910, −12.60993834588362491596291423915, −11.372425919171956572290849921, −10.52829848713340107475277582375, −8.843653469036520792884341951241, −8.432495883481647436295137726384, −6.80575232396162645634085031859, −5.79078956466476057400669196873, −4.08844150299230655767240661100, −3.11521748476582087482211186579, −2.219339811345658680098254022611, −0.34452252862894909544070815058, 2.93297133619840521440710718218, 4.01342468369070733002253172981, 4.55290598789318300477632907205, 6.06934592401709589755825263342, 7.3639822164079139651255595995, 8.197773208335740734976738732192, 9.14464146418494367579530459966, 10.32573264725996695787685234929, 11.55114521628121257249495274615, 13.12031395134286884592629808584, 13.515031786619168555684467097153, 15.02891976128474136220023960615, 15.53210524125374187379208613373, 16.22703482523384655702589110855, 17.07955567095573504002842272571, 18.52743305912796201877484847602, 19.61514385951165730928144789259, 20.54897040517433446000553537524, 21.41590656333352592740021879807, 22.693131268006116647787224577583, 23.135638511702564314575397525471, 24.064160931021922482071330922796, 25.32089864338991244989127225592, 26.17473672110359872609834319934, 26.52716506460942316543329705205

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