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Properties

Label 1-229-229.29-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.133 - 0.991i$
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  + (−0.614 + 0.789i)2-s + (−0.191 + 0.981i)3-s + (−0.245 − 0.969i)4-s + (−0.451 + 0.892i)5-s + (−0.656 − 0.754i)6-s + (0.954 + 0.298i)7-s + (0.915 + 0.401i)8-s + (−0.926 − 0.376i)9-s + (−0.426 − 0.904i)10-s + (0.986 + 0.164i)11-s + (0.998 − 0.0550i)12-s + (−0.837 + 0.546i)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  + (−0.614 + 0.789i)2-s + (−0.191 + 0.981i)3-s + (−0.245 − 0.969i)4-s + (−0.451 + 0.892i)5-s + (−0.656 − 0.754i)6-s + (0.954 + 0.298i)7-s + (0.915 + 0.401i)8-s + (−0.926 − 0.376i)9-s + (−0.426 − 0.904i)10-s + (0.986 + 0.164i)11-s + (0.998 − 0.0550i)12-s + (−0.837 + 0.546i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.133 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.133 - 0.991i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.133 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4214528432 + 0.4820561763i\)
\(L(\frac12)\) \(\approx\) \(-0.4214528432 + 0.4820561763i\)
\(L(1)\) \(\approx\) \(0.3662163498 + 0.5423821720i\)
\(L(1)\) \(\approx\) \(0.3662163498 + 0.5423821720i\)

Euler product

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

−25.156776330590775063492425341457, −24.74688340873157135501457035931, −23.6720897373198865046439705053, −22.70400434609556142227171574854, −21.55884129433254830341227332747, −20.327115393806821050305765366096, −19.96133878763357097429585647155, −18.93657989067353536166830000331, −18.02887245083461589721171167944, −16.93902727345451715028516152723, −16.75750301913232327006985731777, −14.78153341850328614302889823935, −13.63436843960930427823349820120, −12.61512829606089484000858321902, −11.87341846086942103446081090078, −11.25518397563924596797550982368, −9.822979580673988561270644352957, −8.48481145700846717377025649094, −8.00833785175800980794290105852, −6.91902321157512315171992113717, −5.15417363318555889350337329148, −4.03476410240877089984266255365, −2.35771633018156334595729003429, −1.217025509431990649698601479415, −0.29157092554621667502490603441, 1.8671742653561949656133891856, 3.78090975949030615634342848079, 4.80906934179447230062959387292, 5.98791349899653569923060675447, 7.073509716370110218915441251806, 8.226800678952511093024800488477, 9.1918806101224863180358429612, 10.24727289178677130666219449552, 11.08339591139223615612242692431, 11.970054213991162943021124757769, 14.25706923239440095441525166906, 14.70021102305500103154161023635, 15.26679967394429233347638068594, 16.5203326727453206103619993104, 17.25907529353404863451675942249, 18.11438421141537626646519822198, 19.30782356556375751246624419404, 19.98792806679141544265994937743, 21.58555184674069895501542718151, 22.070502341345883343447812236772, 23.341914141565936826769078216299, 23.89612625106125470331606135541, 25.2879781480989733295262176495, 25.959295884767949565516375591310, 27.03097166697323473800555666502

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