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Properties

Label 1-1832-1832.1203-r1-0-0
Degree $1$
Conductor $1832$
Sign $-0.572 - 0.820i$
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  + (−0.926 + 0.376i)3-s + (0.592 − 0.805i)5-s + (−0.821 + 0.569i)7-s + (0.716 − 0.697i)9-s + (0.945 − 0.324i)11-s + (−0.401 − 0.915i)13-s + (−0.245 + 0.969i)15-s + (−0.401 − 0.915i)17-s + (0.993 − 0.110i)19-s + (0.546 − 0.837i)21-s + (0.635 − 0.771i)23-s + (−0.298 − 0.954i)25-s + (−0.401 + 0.915i)27-s + (0.904 + 0.426i)29-s + (−0.754 − 0.656i)31-s + ⋯
L(s)  = 1  + (−0.926 + 0.376i)3-s + (0.592 − 0.805i)5-s + (−0.821 + 0.569i)7-s + (0.716 − 0.697i)9-s + (0.945 − 0.324i)11-s + (−0.401 − 0.915i)13-s + (−0.245 + 0.969i)15-s + (−0.401 − 0.915i)17-s + (0.993 − 0.110i)19-s + (0.546 − 0.837i)21-s + (0.635 − 0.771i)23-s + (−0.298 − 0.954i)25-s + (−0.401 + 0.915i)27-s + (0.904 + 0.426i)29-s + (−0.754 − 0.656i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1832 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1832\)    =    \(2^{3} \cdot 229\)
Sign: $-0.572 - 0.820i$
Analytic conductor: \(196.875\)
Root analytic conductor: \(196.875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1832} (1203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1832,\ (1:\ ),\ -0.572 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7130814664 - 1.367042148i\)
\(L(\frac12)\) \(\approx\) \(0.7130814664 - 1.367042148i\)
\(L(1)\) \(\approx\) \(0.8643017611 - 0.2348663533i\)
\(L(1)\) \(\approx\) \(0.8643017611 - 0.2348663533i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.926 + 0.376i)T \)
5 \( 1 + (0.592 - 0.805i)T \)
7 \( 1 + (-0.821 + 0.569i)T \)
11 \( 1 + (0.945 - 0.324i)T \)
13 \( 1 + (-0.401 - 0.915i)T \)
17 \( 1 + (-0.401 - 0.915i)T \)
19 \( 1 + (0.993 - 0.110i)T \)
23 \( 1 + (0.635 - 0.771i)T \)
29 \( 1 + (0.904 + 0.426i)T \)
31 \( 1 + (-0.754 - 0.656i)T \)
37 \( 1 + (-0.451 - 0.892i)T \)
41 \( 1 + (0.962 - 0.272i)T \)
43 \( 1 + (0.546 - 0.837i)T \)
47 \( 1 + (0.716 - 0.697i)T \)
53 \( 1 + (-0.789 - 0.614i)T \)
59 \( 1 + (-0.451 + 0.892i)T \)
61 \( 1 + (-0.245 - 0.969i)T \)
67 \( 1 + (0.962 + 0.272i)T \)
71 \( 1 + (0.191 + 0.981i)T \)
73 \( 1 + (0.298 + 0.954i)T \)
79 \( 1 + (0.904 - 0.426i)T \)
83 \( 1 + (-0.998 + 0.0550i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.975 - 0.218i)T \)
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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

−19.97759508055256631529514059209, −19.290516167512898068227816745419, −18.87107707368805642344357568925, −17.76660100764060294682756688922, −17.40987416878939228526097993651, −16.73066987581285229063271790389, −15.991285179677098241822011726885, −15.07374609101204607615649868095, −14.05363949178626166182579078374, −13.67684513414177357535485237960, −12.6926581206099378699809928337, −12.065807487214797736193106371273, −11.14926130712997483662820320957, −10.63479953021858403538129037831, −9.64443780800992601286460569599, −9.33415066450716919356633902952, −7.715596141309255615380297527686, −6.9894289311144828168218201869, −6.501351868836822865651981708233, −5.920378914720102704037995558106, −4.795396748264353759275509791365, −3.92449546878260378148833019258, −2.950858840280571094044732135905, −1.76842846186606083624400523023, −1.081592008185148115167421477990, 0.42189571418502462832997428856, 0.89443133252031193337713279564, 2.29484079885122652003257116275, 3.31360208208074274201655575750, 4.32323488338011924866425323508, 5.28542855156802622717290388043, 5.641797199191555299509779852739, 6.54354551492730772198650951453, 7.26872789310047878055596790270, 8.69658224981273801544721712942, 9.28465663802891117143382415476, 9.79663779656981339960131706358, 10.69812106807821512399312748453, 11.58509466659071821952596755886, 12.403699724076150753478068712280, 12.70061092151732549873350059621, 13.700336888548553944142036392724, 14.564022175558351875419351146, 15.71117518413641059930048285038, 16.00480247910018325869486708625, 16.80989898765870839775397491611, 17.400297358779628695362299495148, 18.09641972416013202941737167876, 18.83645138341882931407732239597, 19.93508594444115478506700210181

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