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Properties

Label 1-4033-4033.2653-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.342 - 0.939i$
Analytic cond. $433.406$
Root an. cond. $433.406$
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  + (0.984 − 0.173i)2-s + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (0.396 + 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (0.549 + 0.835i)10-s + (0.549 − 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (−0.0581 + 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.893 + 0.448i)3-s + (0.939 − 0.342i)4-s + (0.396 + 0.918i)5-s + (0.957 + 0.286i)6-s + (−0.993 − 0.116i)7-s + (0.866 − 0.5i)8-s + (0.597 + 0.802i)9-s + (0.549 + 0.835i)10-s + (0.549 − 0.835i)11-s + (0.993 + 0.116i)12-s + (0.116 − 0.993i)13-s + (−0.998 + 0.0581i)14-s + (−0.0581 + 0.998i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.154818056 - 3.078815139i\)
\(L(\frac12)\) \(\approx\) \(2.154818056 - 3.078815139i\)
\(L(1)\) \(\approx\) \(2.308725529 - 0.1126951000i\)
\(L(1)\) \(\approx\) \(2.308725529 - 0.1126951000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (0.893 + 0.448i)T \)
5 \( 1 + (0.396 + 0.918i)T \)
7 \( 1 + (-0.993 - 0.116i)T \)
11 \( 1 + (0.549 - 0.835i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.893 - 0.448i)T \)
31 \( 1 + (-0.597 - 0.802i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.766 + 0.642i)T \)
47 \( 1 + (0.230 - 0.973i)T \)
53 \( 1 + (-0.802 - 0.597i)T \)
59 \( 1 + (-0.998 - 0.0581i)T \)
61 \( 1 + (0.686 + 0.727i)T \)
67 \( 1 + (0.918 + 0.396i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.835 - 0.549i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (-0.686 - 0.727i)T \)
97 \( 1 + (0.396 + 0.918i)T \)
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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

−18.626505081399964656660189877532, −17.712846098651021522684141276131, −17.00156046555011166622703461870, −16.26985188747891665472970494162, −15.66957292951027618161229343010, −15.07851919151283853696281379478, −14.13304555114642634230504282032, −13.80356634973358014841895453174, −12.98784179837610423287816305206, −12.61997289477447371533855682866, −12.08196879610094712435529503242, −11.20626183751498436212522328069, −9.98401552431009752208599850333, −9.373984911161459255789243195655, −8.852942747608934433010691121572, −7.94832487557005984408099972638, −7.060916725113604231221117555316, −6.584053575196440683568172385258, −5.922095127712144520884459660019, −4.88498595589331247295941364983, −4.14210400623536823129831660769, −3.61428715894343152645692315744, −2.6377679678527721655550253371, −1.822521956719281918942898600871, −1.37955650791340184576734605970, 0.25179832398624235183178871015, 1.64134215665362313254967701309, 2.47066824264262344170332107511, 3.13008902744087690710393371760, 3.595535009921497164020677963143, 4.1876887608591357936772330262, 5.53159805599708140782366783055, 5.87373049953672634596204712228, 6.805543241603382515857852845248, 7.40442565819476194295412647270, 8.22569016750608836890984102589, 9.37540805163053405439520779406, 9.877012415498265002968887112028, 10.47473691946724876305581876690, 11.18020959151737784565797229789, 11.92863446191861544434144914015, 13.00640064479159499125842489091, 13.43104554006420277970274788304, 13.96118961977362504740465162457, 14.59786944489291718689524712484, 15.18088586313862164485073153757, 15.97400813202432643231966012453, 16.26725552779119490455474857725, 17.26254068335027729368531465899, 18.53303971758530667907142280167

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