↑

Properties

Label 1-4033-4033.34-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.232 - 0.972i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.766 + 0.642i)2-s + 3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + 9-s + (0.939 + 0.342i)10-s + (0.939 − 0.342i)11-s + (0.173 − 0.984i)12-s − 13-s + (0.939 + 0.342i)14-s + (−0.5 − 0.866i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + 3-s + (0.173 − 0.984i)4-s + (−0.5 − 0.866i)5-s + (−0.766 + 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + 9-s + (0.939 + 0.342i)10-s + (0.939 − 0.342i)11-s + (0.173 − 0.984i)12-s − 13-s + (0.939 + 0.342i)14-s + (−0.5 − 0.866i)15-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.232 - 0.972i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.232 - 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232575826 - 0.9726362613i\)
\(L(\frac12)\) \(\approx\) \(1.232575826 - 0.9726362613i\)
\(L(1)\) \(\approx\) \(0.9884018918 - 0.1596840937i\)
\(L(1)\) \(\approx\) \(0.9884018918 - 0.1596840937i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−18.63681502580040490625648278212, −18.3334963717228297379928995206, −17.46099949137506528745389886971, −16.566430090211059931074856450573, −15.89051586861128145682391896086, −15.12174385593416153081505416341, −14.64368389469283995993127479974, −13.98371627640832452495720389404, −12.7719349996121594351638918371, −12.462683019449671853287321848600, −11.79922936344005676904474157706, −10.90996942270831997082165443717, −10.1944713264767959657209688196, −9.492841115389889499336579370445, −9.10196947418374801091519518806, −8.21256859277034220587201631912, −7.66330676714327733564200343899, −6.85344198112263010453193049844, −6.39343093164503261013183430092, −4.867605407163021455732724500197, −3.8614061339252767091877579261, −3.435393101303115550971529981087, −2.54340697685633234477903139327, −2.19533941643213417288553271993, −1.07420275223250678707309987287, 0.61220372585155122395453283407, 1.14839534360289367489342887728, 2.238574374460450360522473208069, 3.26491628345878595829417203307, 4.07727433981391196260335011686, 4.81337832182086393197246616674, 5.596037967578289457174624527294, 6.89715927256029368583323684857, 7.24868310971834790724266796175, 7.73416647448435090358720284768, 8.74318813142288841188616648031, 9.19562914695056863871838121204, 9.661662849734877858580607671602, 10.39556643514235437872221398650, 11.468805410036397013626451131511, 12.07119588213835644081496107813, 13.12831947088036697743417627119, 13.77809282939607309941463226994, 14.204482823788042754066083604508, 15.09299387604878878466209848218, 15.73207281028176252010682525027, 16.27575534317155621334257927593, 16.87525951909308682516541832435, 17.49102314964691069891581203041, 18.35077013621556936621423143122

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