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Properties

Label 1-4033-4033.100-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0192 - 0.999i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
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  + (−0.173 − 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 + 0.342i)4-s + (0.597 − 0.802i)5-s + (−0.973 + 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.893 − 0.448i)10-s + (−0.893 + 0.448i)11-s + (0.396 + 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 + 0.342i)4-s + (0.597 − 0.802i)5-s + (−0.973 + 0.230i)6-s + (0.396 + 0.918i)7-s + (0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.893 − 0.448i)10-s + (−0.893 + 0.448i)11-s + (0.396 + 0.918i)12-s + (−0.396 − 0.918i)13-s + (0.835 − 0.549i)14-s + (−0.835 − 0.549i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0192 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0192 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0192 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0192 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6801893885 - 0.6934112558i\)
\(L(\frac12)\) \(\approx\) \(0.6801893885 - 0.6934112558i\)
\(L(1)\) \(\approx\) \(0.6004788823 - 0.5335833634i\)
\(L(1)\) \(\approx\) \(0.6004788823 - 0.5335833634i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.0581 - 0.998i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (-0.893 + 0.448i)T \)
13 \( 1 + (-0.396 - 0.918i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (-0.0581 - 0.998i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 - T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.686 - 0.727i)T \)
53 \( 1 + (0.993 + 0.116i)T \)
59 \( 1 + (0.835 + 0.549i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (-0.597 - 0.802i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (-0.396 + 0.918i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (-0.286 + 0.957i)T \)
97 \( 1 + (0.597 - 0.802i)T \)
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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

−18.38044457007958236546086854665, −17.79735502327393019687337576659, −17.14123939826860401099773819275, −16.57771928083245010145573412051, −16.00688457784882779784352231662, −15.25490214050083872091072477642, −14.51832000921689719963479922039, −14.19636582250609665778686957078, −13.541085834400382179640568504652, −12.803374090906553406518851885608, −11.32934162235004812858567797854, −10.84781328326269417844624127009, −10.330316035118211506115417997922, −9.61446605771433132352944663013, −8.921181167565205497749070479048, −8.28092100381127812716231126837, −7.197165744955309532145291218047, −6.83854110210773199816480441058, −5.97155660022838059718650143104, −5.15412752233858820354694903557, −4.57775568104608873407916546360, −3.9109462560096046318215546001, −2.90528577559590623886950016733, −2.00064424835857720218922459376, −0.45274250298151768620878351695, 0.60685986668065916117159114817, 1.86321948583623462668869468977, 1.995294457935626766056066691450, 2.74782317608280970807158276275, 3.8984147761954642500111553709, 4.94231365847697137019689114168, 5.52867490565555403013121492564, 5.989874835267674922862842291718, 7.3892249839286594937419809721, 8.04848828539207444977463360139, 8.597017258723606121034414215626, 9.16997070942044809862759834568, 10.136737284237599554651200440028, 10.69520757566874642462467442042, 11.652504381989642420308200013346, 12.29119387920194911314002491537, 12.6749395533955089200397167258, 13.29744544551130931727326694126, 13.75727218859363512510493396162, 14.85925868487052054158198815826, 15.373516081664486802735039681867, 16.65735205962160189770561110568, 17.30685414339162797843515365287, 17.80804511628442971219161395046, 18.23604436346638306789966670681

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