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Properties

Label 1-4033-4033.2669-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.0463 + 0.998i$
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 + (0.286 + 0.957i)3-s + 4-s + (0.448 + 0.893i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (0.286 + 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (−0.727 + 0.686i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.286 + 0.957i)3-s + 4-s + (0.448 + 0.893i)5-s + (−0.286 − 0.957i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.448 − 0.893i)10-s + (0.448 − 0.893i)11-s + (0.286 + 0.957i)12-s + (−0.893 + 0.448i)13-s + (0.835 + 0.549i)14-s + (−0.727 + 0.686i)15-s + 16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0463 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.0463 + 0.998i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.0463 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5665460261 + 0.5934351091i\)
\(L(\frac12)\) \(\approx\) \(0.5665460261 + 0.5934351091i\)
\(L(1)\) \(\approx\) \(0.6196863850 + 0.2544659692i\)
\(L(1)\) \(\approx\) \(0.6196863850 + 0.2544659692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.286 + 0.957i)T \)
5 \( 1 + (0.448 + 0.893i)T \)
7 \( 1 + (-0.835 - 0.549i)T \)
11 \( 1 + (0.448 - 0.893i)T \)
13 \( 1 + (-0.893 + 0.448i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (0.549 - 0.835i)T \)
31 \( 1 + (0.802 + 0.597i)T \)
41 \( 1 + (-0.342 - 0.939i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.957 + 0.286i)T \)
53 \( 1 + (0.230 - 0.973i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (0.918 + 0.396i)T \)
67 \( 1 + (0.957 + 0.286i)T \)
71 \( 1 - T \)
73 \( 1 + (0.835 - 0.549i)T \)
79 \( 1 + (0.686 + 0.727i)T \)
83 \( 1 + (0.993 - 0.116i)T \)
89 \( 1 + (0.998 + 0.0581i)T \)
97 \( 1 + (-0.802 + 0.597i)T \)
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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

−18.1087808893120536417205011344, −17.63179094165919300473001986494, −17.201171686058840037171043544196, −16.54125035415617291315675184287, −15.568251606659312522052180461289, −15.098334198784903551890233724095, −14.270119034672590740236345800199, −13.1930095881994130450134017881, −12.668185897393415091240034591437, −12.25554492506381897894187686773, −11.62949726030647229004168404914, −10.46310675925441947584057598931, −9.70728197412383083506582050877, −9.26890892738762561820392602508, −8.538628881087051197351950372577, −8.01559528236196576591103075498, −7.12563518299471754785435434953, −6.4136687034639950508260127749, −6.00372067568885348343696178348, −4.98013084898296055287940799998, −3.82732905482363230890482653876, −2.58909045852899516028996651273, −2.25222907265748501502146919245, −1.439522878465478321973181745315, −0.45919629287857056523763106014, 0.643002446078703396826303341501, 2.28926200077625696617570026119, 2.51566768807339306966734866662, 3.54498338553143221473197988853, 4.07221453532270903501579678162, 5.375513824176802400691719598002, 6.25216468000985731648907486068, 6.68563345893209252003831366087, 7.51677689677652528441429665797, 8.39375706328852671489361953767, 9.1347889145567792855621331037, 9.69162025113969413935935158892, 10.39000827429930501544114733794, 10.57105807458865443737165816228, 11.59279180705767963176443760310, 12.09944088803994744424509046973, 13.45702176589555812814720886018, 14.120260851345967675092288640894, 14.54617300915157601350735957927, 15.52579178820964853608529941794, 15.98063060930822407920719189824, 16.69045921436951563140256552475, 17.185157515706750937160486744973, 17.82614084217897358920152623610, 18.91822659089807699550883109834

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