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Properties

Label 1-4033-4033.108-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.556 - 0.831i$
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 + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (0.766 + 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.556 - 0.831i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.556 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.078667060 - 1.110299801i\)
\(L(\frac12)\) \(\approx\) \(2.078667060 - 1.110299801i\)
\(L(1)\) \(\approx\) \(1.182935786 - 0.6122396411i\)
\(L(1)\) \(\approx\) \(1.182935786 - 0.6122396411i\)

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

−18.356258195391175006211098607720, −17.41239685649868804969507452807, −17.230031316898561001842931982928, −16.46332349744837547134485405454, −16.03310395934702943093420583665, −15.313728612551252548878942575238, −14.316126217232868941823463700715, −14.05609701163273860157654552106, −13.559070078508307267001697138886, −12.68163640678262982164935323411, −11.348102197449442996806997418832, −10.955893857155870056433725180560, −10.060219289198859257643806635511, −9.30416706496551178281919535414, −8.93839325691404046798058736239, −8.25485884031975348313804698813, −7.53228537264528357406564538670, −6.46458579000463425834596660142, −5.77857684360512124646567749810, −5.16926164254274188844399382440, −4.51376504018466971494584554634, −3.85636751570624496473008971080, −2.948496856292276939867805921508, −1.42545200520316079122474637498, −0.85763454106737916746451515420, 1.055253084748776376243478470044, 1.629115967052455009111565581744, 2.27031751957162078581638205355, 2.93503326168466088089225420118, 3.81358334271035473145138937228, 4.80793600264871266812994309365, 5.95849483273873775360929105414, 6.06092852729078744397676594362, 7.45570561536192626216484083581, 7.89225757752412668554249393307, 8.77476096140824919631680107700, 9.28184005060192275772981266430, 10.249511496237653135843633011783, 10.788033155470892278475463447685, 11.59228860561489315622502994618, 12.26727226706142790234521058969, 12.66509700602017985383725760086, 13.55115360348358817080757182527, 14.16375608759318479001229516752, 14.62243009233770286730095338243, 15.31094008044505126551150529841, 16.92561872037794176032965104098, 17.21026464399678209169232308736, 18.023078718286139465086195479913, 18.414389614417021307749374767425

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