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Properties

Label 1-4033-4033.3565-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.808 - 0.588i$
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 + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.866 − 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s − i·15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (−0.866 − 0.5i)5-s + (−0.766 + 0.642i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (−0.5 − 0.866i)13-s + (−0.939 − 0.342i)14-s − i·15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.808 - 0.588i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (3565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.808 - 0.588i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3103562852 + 0.9540741376i\)
\(L(\frac12)\) \(\approx\) \(-0.3103562852 + 0.9540741376i\)
\(L(1)\) \(\approx\) \(0.5962498600 + 0.6910142165i\)
\(L(1)\) \(\approx\) \(0.5962498600 + 0.6910142165i\)

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

−18.275966016719038161001765742523, −17.74210393589703209305213351996, −16.8516492570484818556628748792, −15.998608801445598772286779466828, −14.95325849012984740612326044644, −14.48986062003263717408143741771, −13.87523217164004340754847575843, −13.21087436520782982383464279950, −12.438921886020685010706275237769, −12.02052354316333745495550351546, −11.37219776223937642727741649869, −10.49200301203097588320470963860, −9.830697082616023848378009971459, −9.19786169123387167493387547996, −8.25430962593254034111042044848, −7.48302251737614524011700453764, −7.04567383831847574538739300822, −6.27430376242200132933774528029, −4.90810445557136919573770170599, −4.37610116566359537208310798270, −3.39819288421049246475558909739, −2.963269362994158262781429026781, −2.17702001649951669269254461925, −1.15887053009357579574577987102, −0.32733974772040551014184084619, 0.95610039667689783755904514145, 2.71220264597473490139872704098, 3.41622139092226488110883906325, 3.75477715292268241305349577883, 4.91258899853393102728590673446, 5.514789652845737348478933742025, 5.78645862979674546047424892283, 7.24423166249383118840425817915, 7.79707631705693595004520764308, 8.50066545068958522098401118799, 8.85396260617955872908168118382, 9.76683500122636110639883579831, 10.276189075338485047330602678050, 11.47931372016742751181295399291, 12.09957605212041778507275927198, 12.961839456677862709853550690225, 13.42381572534961478907887294354, 14.54016513682072682204275348043, 14.84491225211664986865837880090, 15.64097418383799517471159107385, 15.998907743132259808875587375011, 16.544964968191943311628632563612, 17.13777479955845853924833000694, 18.25318516351396859480368766101, 18.8769760305440334381995004570

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