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Properties

Label 1-229-229.22-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.930 - 0.365i$
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 + (0.945 − 0.324i)3-s + (−0.245 − 0.969i)4-s + (−0.546 − 0.837i)5-s + (0.324 − 0.945i)6-s + (0.735 − 0.677i)7-s + (−0.915 − 0.401i)8-s + (0.789 − 0.614i)9-s + (−0.996 − 0.0825i)10-s + (0.986 + 0.164i)11-s + (−0.546 − 0.837i)12-s + (0.837 − 0.546i)13-s + (−0.0825 − 0.996i)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 + (0.945 − 0.324i)3-s + (−0.245 − 0.969i)4-s + (−0.546 − 0.837i)5-s + (0.324 − 0.945i)6-s + (0.735 − 0.677i)7-s + (−0.915 − 0.401i)8-s + (0.789 − 0.614i)9-s + (−0.996 − 0.0825i)10-s + (0.986 + 0.164i)11-s + (−0.546 − 0.837i)12-s + (0.837 − 0.546i)13-s + (−0.0825 − 0.996i)14-s + (−0.789 − 0.614i)15-s + (−0.879 + 0.475i)16-s + (0.546 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.930 - 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6983256800 - 3.686414959i\)
\(L(\frac12)\) \(\approx\) \(0.6983256800 - 3.686414959i\)
\(L(1)\) \(\approx\) \(1.325260209 - 1.571497185i\)
\(L(1)\) \(\approx\) \(1.325260209 - 1.571497185i\)

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 \( 1 + (0.945 - 0.324i)T \)
5 \( 1 + (-0.546 - 0.837i)T \)
7 \( 1 + (0.735 - 0.677i)T \)
11 \( 1 + (0.986 + 0.164i)T \)
13 \( 1 + (0.837 - 0.546i)T \)
17 \( 1 + (0.546 + 0.837i)T \)
19 \( 1 + (0.546 - 0.837i)T \)
23 \( 1 + (-0.996 + 0.0825i)T \)
29 \( 1 + (-0.735 + 0.677i)T \)
31 \( 1 + (-0.164 + 0.986i)T \)
37 \( 1 + (-0.879 - 0.475i)T \)
41 \( 1 + (-0.614 + 0.789i)T \)
43 \( 1 + (-0.879 - 0.475i)T \)
47 \( 1 + (0.614 + 0.789i)T \)
53 \( 1 + (0.945 - 0.324i)T \)
59 \( 1 + (0.475 + 0.879i)T \)
61 \( 1 + (0.789 - 0.614i)T \)
67 \( 1 + (0.614 + 0.789i)T \)
71 \( 1 + (0.986 - 0.164i)T \)
73 \( 1 + (0.915 + 0.401i)T \)
79 \( 1 + (-0.735 - 0.677i)T \)
83 \( 1 + (-0.879 + 0.475i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.401 + 0.915i)T \)
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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

−26.36898294221623403889828261470, −25.5579534999849699521969315407, −24.74690692865486533202916678014, −23.97644625509057845877099142378, −22.71608285634270282432090873721, −22.049536250389949542685630565710, −21.0946997835425549170580872096, −20.27506688125594125041689265086, −18.76040343594886966798922353992, −18.35378088502660181104468818272, −16.75582363381928634452397930044, −15.78022945989149627536111350979, −15.0509157506100224960813578197, −14.216567666548098905589051746932, −13.77373723382384459983227932330, −12.05727669379593738833537256815, −11.4161741387893112995611597568, −9.71745326315519545419976002471, −8.57447558724299762251027504411, −7.85609865178377822618731325178, −6.816476342287124017228595323, −5.5573059655227872923531071779, −4.08794074621974456722302409942, −3.48674351451583502838529192656, −2.105249795409076575681458308906, 0.99314182253582732530130007195, 1.68011810780745770238686785102, 3.52445140315355209299337163643, 4.03614201600572887250910127065, 5.312799490268040406611056458639, 6.925941587276886542246230671596, 8.21517263221918107250965181580, 9.00389792750985601460451709108, 10.21793587994124069708333869852, 11.4358601630122642480134010271, 12.35535553315515017728689597000, 13.24655033523551422439499365459, 14.09366721098991259668266363346, 14.91683451953057538235914235584, 15.9443727539384874643310373812, 17.44437541234093530156484221026, 18.50328411469630664389243922473, 19.800436142372053815302097112094, 19.98979708786888966547175342388, 20.82383894072114569871044430491, 21.74018910459864240032561832116, 23.11950085294710665567408288567, 23.938268863328340558978402165736, 24.449337898043887378401465316663, 25.580718068148825714268514895852

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