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Properties

Label 1-4033-4033.1058-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.339 - 0.940i$
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 + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + (−0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.984 − 0.173i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + (−0.766 + 0.642i)6-s + (0.766 + 0.642i)7-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)13-s + (0.766 − 0.642i)14-s + (0.984 − 0.173i)15-s + (0.766 + 0.642i)16-s + (0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.339 - 0.940i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.339 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4633652618 - 0.6598410792i\)
\(L(\frac12)\) \(\approx\) \(0.4633652618 - 0.6598410792i\)
\(L(1)\) \(\approx\) \(0.6200473356 - 0.3533639500i\)
\(L(1)\) \(\approx\) \(0.6200473356 - 0.3533639500i\)

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 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.642 + 0.766i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 - T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (-0.342 - 0.939i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.984 + 0.173i)T \)
53 \( 1 + iT \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.642 + 0.766i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.984 + 0.173i)T \)
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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

−18.45907381684676182211325856561, −17.74608684984009579138246037463, −16.96691554014688468544778468984, −16.567310621405639636129645932789, −16.13464533206050405903556669390, −15.52264801140949051226297331699, −14.58880944157493226778794176108, −14.20069072546420616609361780571, −13.28531917529477103219159335511, −12.41703308269650930280957765157, −11.99565785831749700647531741460, −11.0392923353816566139474450406, −10.47235799822769332634266700772, −9.58492151791361456445134272082, −8.68035368194951613984451215657, −8.267576007956845104239272324423, −7.507055832260341909312260651220, −6.659043522068068293544639600117, −5.878913311703990658017245586950, −5.13710161277739068496621425023, −4.6780249631381759412159772443, −3.86368853868167748750774416666, −3.5087305829114702842313276292, −1.59156449370472370202918343221, −0.63899088895715459732765582186, 0.40368179325153647196468434555, 1.47706697071680801400493776532, 2.49423146730113604636064650132, 2.688870356619672191535572206987, 4.099483047777621689877274919264, 4.589639143153257029338272130916, 5.667014887959918062661245722503, 5.879444995645498003744920894068, 7.16917500636590364201362402277, 7.99678689521538037888748106983, 8.16390662806023956393601311844, 9.52055208318796700619865997213, 10.40636503258079657860244864202, 10.722282399031483233803706127211, 11.49577496348829799190489158936, 12.09211127568685660829310151967, 12.45685935536509690624105696714, 13.29539036450516737927844420808, 14.003731362913965574794937343664, 14.95094928484862572385364588390, 15.23133383536582033179559953729, 16.22385670684590002854209974723, 17.397856575605676752034258726542, 17.77421138223374423729441928545, 18.42924687029470165493279364883

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