Properties

Label 1-4033-4033.27-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.0287 - 0.999i$
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  − 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.766 − 0.642i)5-s + (0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.766 − 0.642i)13-s + (−0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  − 2-s + (−0.939 + 0.342i)3-s + 4-s + (−0.766 − 0.642i)5-s + (0.939 − 0.342i)6-s + (0.766 + 0.642i)7-s − 8-s + (0.766 − 0.642i)9-s + (0.766 + 0.642i)10-s + (0.766 − 0.642i)11-s + (−0.939 + 0.342i)12-s + (−0.766 − 0.642i)13-s + (−0.766 − 0.642i)14-s + (0.939 + 0.342i)15-s + 16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.0287 - 0.999i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.0287 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3063161789 - 0.2976256270i\)
\(L(\frac12)\) \(\approx\) \(0.3063161789 - 0.2976256270i\)
\(L(1)\) \(\approx\) \(0.4703910118 + 0.02360871187i\)
\(L(1)\) \(\approx\) \(0.4703910118 + 0.02360871187i\)

Euler product

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

−18.37337749165793268190581108058, −17.96977466063305701046879608136, −17.348133400906898343442573605163, −16.842274796349269474404522593530, −16.11533199137701845854475239831, −15.425875221180149962923559732636, −14.6611623543411859712736826083, −14.118085870248082407511728582590, −12.822116404406622408789874943074, −12.20314274184972517159374959851, −11.41256831233410356210672895589, −11.1680781462209464428709629583, −10.56980765094673066706676332963, −9.719852113546352171151225362511, −8.9350685354731163495564633800, −8.004365996604084729151161722518, −7.295802960898053284402658679851, −6.80172389244041504571213851196, −6.57187969309985158249901897844, −5.070805833950063233749804981161, −4.566339190671942300372404521578, −3.60199938143789880276203884156, −2.38966766434803092390897719403, −1.70928636197654436944652689712, −0.77812242775102944004686662942, 0.281023728709558089412878251544, 1.2012214834603798561454611996, 1.9649896067241129986932285022, 3.239045090642399273910037266044, 4.037157237896360176385089279920, 4.989555336806438115276431840742, 5.622037351606406429634222285380, 6.28596718406781985140341214527, 7.344126992031117893911615191773, 7.82697472304183921214419426695, 8.65751052167575565626818330024, 9.28934208576212879572299622319, 9.879704120267705607345156258434, 10.94933705885491775412496390272, 11.384718071296466613710963952388, 11.83848065824546704376225024179, 12.40038277563287630105846344421, 13.29088437367703262200254119193, 14.68279213734895592615251718398, 15.249144613265696903976760071033, 15.62271517556229446780175654200, 16.59820765786987394182006715072, 16.85826682100661470057281604217, 17.58149474836072229029792075286, 18.145582003140634008748275057616

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