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Properties

Label 1-4033-4033.1086-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.835 - 0.549i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.984 − 0.173i)2-s + (0.5 − 0.866i)3-s + (0.939 − 0.342i)4-s + i·5-s + (0.342 − 0.939i)6-s + 7-s + (0.866 − 0.5i)8-s + (−0.5 − 0.866i)9-s + (0.173 + 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.866 − 0.5i)13-s + (0.984 − 0.173i)14-s + (0.866 + 0.5i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.5 − 0.866i)3-s + (0.939 − 0.342i)4-s + i·5-s + (0.342 − 0.939i)6-s + 7-s + (0.866 − 0.5i)8-s + (−0.5 − 0.866i)9-s + (0.173 + 0.984i)10-s + (−0.173 + 0.984i)11-s + (0.173 − 0.984i)12-s + (−0.866 − 0.5i)13-s + (0.984 − 0.173i)14-s + (0.866 + 0.5i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.835 - 0.549i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.835 - 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.200336996 - 4.010022300i\)
\(L(\frac12)\) \(\approx\) \(1.200336996 - 4.010022300i\)
\(L(1)\) \(\approx\) \(2.063722755 - 0.8222338647i\)
\(L(1)\) \(\approx\) \(2.063722755 - 0.8222338647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + iT \)
7 \( 1 + T \)
11 \( 1 + (-0.173 + 0.984i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.342 + 0.939i)T \)
31 \( 1 + (-0.984 - 0.173i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.766 - 0.642i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.173 - 0.984i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (0.342 - 0.939i)T \)
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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

−18.880494076546555762551830050448, −17.29012488300336620002093465314, −17.0842654261430950410460481247, −16.49940010516022107554498190112, −15.61244843395968378025776361269, −15.206946268269602331057451771266, −14.41668333223661045170405490216, −14.00262133076418578224236540379, −13.14998530927525230121585620302, −12.65451939669063028486295450852, −11.59202169544940860661160143427, −11.18841669065222070210798956417, −10.51710484694656337259492639968, −9.42375433230712500736818478090, −8.81368750976392322112801667110, −7.99645761747356501494019404534, −7.67448828814034555400094825721, −6.37942693540783341715960017412, −5.45118239374694264730130226800, −5.06803135406874034271488535836, −4.35404430868711468012343041182, −3.84103099281556249614528523435, −2.84347682495589190673947413824, −2.06330016975386160842813447600, −1.219764435661767073328656860845, 0.311476785890358565116127613262, 1.67268818874682871183854192360, 2.047210637427756633670647752808, 2.84639298551338870071338429852, 3.4675510341720798698233032788, 4.4987628832831658081695413616, 5.23680299297479436053594179904, 5.98672542620688111572898641374, 7.00616747421096519342251513611, 7.33183569411377629043940197692, 7.78373667037890788080782031736, 8.929748673469423183918560371248, 9.937797669518454542615917081161, 10.64566586544601801151093163104, 11.25668781435458592001729493229, 12.20854399504418962468750452117, 12.38610025512864330524109787007, 13.29569729527588938335172216112, 14.128202302208936258786707683764, 14.51221743728519260235818288138, 14.974909895663833098635120959693, 15.46316144333354149584457449635, 16.77906953066475838744055165165, 17.41189351836482483201114663096, 18.29095901563587337791901931198

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