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Properties

Label 1-4033-4033.2153-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.469 + 0.882i$
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 + (0.173 + 0.984i)5-s + (0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (−0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.939 + 0.342i)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 + (0.173 + 0.984i)5-s + (0.939 − 0.342i)6-s + (0.173 + 0.984i)7-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.939 − 0.342i)10-s + (0.939 + 0.342i)11-s + (−0.5 − 0.866i)12-s + (0.939 + 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.939 + 0.342i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.469 + 0.882i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.469 + 0.882i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9154007107 + 1.523994562i\)
\(L(\frac12)\) \(\approx\) \(0.9154007107 + 1.523994562i\)
\(L(1)\) \(\approx\) \(1.045295347 + 0.3748491570i\)
\(L(1)\) \(\approx\) \(1.045295347 + 0.3748491570i\)

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

−18.24717330885119848476363078207, −17.32269028344258612816880606527, −17.01606845157751077628615365531, −16.3298985163983786077372136140, −15.792253071494718754219117283668, −14.56356798014279568712043703660, −14.101596055304997045895735103718, −13.59162593030303931969538166566, −13.0993111998155124412643783105, −12.18127058144217892226185425246, −11.637865921492765884866248382485, −10.47744904802334699606565012864, −9.67810216142806681408724329046, −8.97274818107934787216263337821, −8.267193276017587279424315020662, −7.828552195725833414158459157100, −7.089337324224755860845494608668, −6.34214135944947664152346516689, −5.658840618775858699075750521317, −5.07300092483917183388942259074, −3.82556080215128579434311996130, −3.560206077830494146789951813908, −1.79478216491104784758533194945, −1.0786906362359573931132316829, −0.62619553686265453055604010824, 1.30538334056142165454470024614, 2.12988918671434356406939322678, 2.957005197310531138175016687695, 3.51132294280942616999142415546, 4.1592631091723085440395266751, 5.10168705114903443340755508193, 5.845179451226267141836749718665, 6.5938617447901543002191342593, 7.96190267276090496144284822043, 8.38717177862290293777181029672, 9.373701296186589164024882828644, 9.70712440237933768385777662578, 10.32101327715970515925407704656, 11.25376874481207115019302835043, 11.61128896902871616994917574706, 12.1386685844426456057084102581, 13.31752196518644443626126261993, 14.07433396021947180251712511673, 14.50442142714536917504128306540, 15.11589228871605245167223397270, 16.0088721747783843620186618659, 16.621101182655568761050260066855, 17.79702746230824694818364779052, 17.880416690462274904988347206443, 18.80844650629805479264703048582

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