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Properties

Label 1-1832-1832.1019-r1-0-0
Degree $1$
Conductor $1832$
Sign $0.972 - 0.232i$
Analytic cond. $196.875$
Root an. cond. $196.875$
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  + (−0.191 + 0.981i)3-s + (−0.451 + 0.892i)5-s + (−0.298 + 0.954i)7-s + (−0.926 − 0.376i)9-s + (−0.986 − 0.164i)11-s + (0.546 + 0.837i)13-s + (−0.789 − 0.614i)15-s + (0.546 + 0.837i)17-s + (−0.998 − 0.0550i)19-s + (−0.879 − 0.475i)21-s + (0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (0.546 − 0.837i)27-s + (0.975 − 0.218i)29-s + (0.350 + 0.936i)31-s + ⋯
L(s)  = 1  + (−0.191 + 0.981i)3-s + (−0.451 + 0.892i)5-s + (−0.298 + 0.954i)7-s + (−0.926 − 0.376i)9-s + (−0.986 − 0.164i)11-s + (0.546 + 0.837i)13-s + (−0.789 − 0.614i)15-s + (0.546 + 0.837i)17-s + (−0.998 − 0.0550i)19-s + (−0.879 − 0.475i)21-s + (0.904 + 0.426i)23-s + (−0.592 − 0.805i)25-s + (0.546 − 0.837i)27-s + (0.975 − 0.218i)29-s + (0.350 + 0.936i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.972 - 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $0.972 - 0.232i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ 0.972 - 0.232i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04343858285 + 0.005117408272i\)
\(L(\frac12)\) \(\approx\) \(-0.04343858285 + 0.005117408272i\)
\(L(1)\) \(\approx\) \(0.5264674077 + 0.4678486558i\)
\(L(1)\) \(\approx\) \(0.5264674077 + 0.4678486558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.191 + 0.981i)T \)
5 \( 1 + (-0.451 + 0.892i)T \)
7 \( 1 + (-0.298 + 0.954i)T \)
11 \( 1 + (-0.986 - 0.164i)T \)
13 \( 1 + (0.546 + 0.837i)T \)
17 \( 1 + (0.546 + 0.837i)T \)
19 \( 1 + (-0.998 - 0.0550i)T \)
23 \( 1 + (0.904 + 0.426i)T \)
29 \( 1 + (0.975 - 0.218i)T \)
31 \( 1 + (0.350 + 0.936i)T \)
37 \( 1 + (-0.851 + 0.523i)T \)
41 \( 1 + (-0.137 + 0.990i)T \)
43 \( 1 + (-0.879 - 0.475i)T \)
47 \( 1 + (-0.926 - 0.376i)T \)
53 \( 1 + (-0.945 + 0.324i)T \)
59 \( 1 + (-0.851 - 0.523i)T \)
61 \( 1 + (-0.789 + 0.614i)T \)
67 \( 1 + (-0.137 - 0.990i)T \)
71 \( 1 + (-0.635 - 0.771i)T \)
73 \( 1 + (0.592 + 0.805i)T \)
79 \( 1 + (0.975 + 0.218i)T \)
83 \( 1 + (0.0275 - 0.999i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
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

−19.24748054183211744460663255432, −18.62964122144663577544754634036, −17.70525989254196884530480900342, −17.1482075532390465009906014300, −16.39316310788976471636978360121, −15.78349615415698595999389826365, −14.800619462199245764642458354709, −13.72613670496453829131393339923, −13.22154542009347711002955667494, −12.67485744736982140442660104238, −12.027729738452758718529857709064, −10.968238235381224773870717458930, −10.475726917717367249299187892840, −9.345288925444173337372030471687, −8.28478838301209760099533556896, −7.92494750971215901443811570894, −7.12031550188777813997513230182, −6.29985636508219958309954985045, −5.27856803064696243487902209383, −4.68855490349728295928109885407, −3.49409126030072920424205254976, −2.659165307706231292912903599564, −1.40667830457350156041753906478, −0.61388904191081146470147999443, −0.01255190513115796218015868888, 1.79672627687006603358584040774, 3.032235770365343951849907140451, 3.276061047805815256739070100260, 4.4566630728755995397528024487, 5.19627541973796679629784528924, 6.232951083473242451946490522419, 6.61994504819414996566678289076, 8.06778513418274858518455826615, 8.55551720383190382736411904380, 9.46185238805013612313453952415, 10.40167147064125805548743339876, 10.76456868155513251359428121766, 11.65658830418577822113448230761, 12.25657050030651763697468198560, 13.32774211263557414839326950173, 14.26127150980816463167931271955, 15.05066850983766435785948530223, 15.45003449779647884102888698959, 16.072920273135699490868911788897, 16.87604144513539771328854926748, 17.77783591147764745407420542111, 18.61741746518971584742908150749, 19.12198323393751513669183074956, 19.81811204746109958374866778032, 21.114305350501674678198316020629

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