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Properties

Label 1-4033-4033.1038-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.995 + 0.0957i$
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  + (0.939 + 0.342i)2-s + (0.686 − 0.727i)3-s + (0.766 + 0.642i)4-s + (0.998 + 0.0581i)5-s + (0.893 − 0.448i)6-s + (0.893 − 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.993 − 0.116i)12-s + (0.835 + 0.549i)13-s + (0.993 − 0.116i)14-s + (0.727 − 0.686i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.686 − 0.727i)3-s + (0.766 + 0.642i)4-s + (0.998 + 0.0581i)5-s + (0.893 − 0.448i)6-s + (0.893 − 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.0581 − 0.998i)9-s + (0.918 + 0.396i)10-s + (−0.918 + 0.396i)11-s + (0.993 − 0.116i)12-s + (0.835 + 0.549i)13-s + (0.993 − 0.116i)14-s + (0.727 − 0.686i)15-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.995 + 0.0957i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.995 + 0.0957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.988895994 + 0.2874009153i\)
\(L(\frac12)\) \(\approx\) \(5.988895994 + 0.2874009153i\)
\(L(1)\) \(\approx\) \(2.976092802 + 0.09179008714i\)
\(L(1)\) \(\approx\) \(2.976092802 + 0.09179008714i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.939 + 0.342i)T \)
3 \( 1 + (0.686 - 0.727i)T \)
5 \( 1 + (0.998 + 0.0581i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.918 + 0.396i)T \)
13 \( 1 + (0.835 + 0.549i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.918 - 0.396i)T \)
31 \( 1 + (-0.230 - 0.973i)T \)
41 \( 1 + (0.984 - 0.173i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (-0.549 - 0.835i)T \)
53 \( 1 + (-0.116 - 0.993i)T \)
59 \( 1 + (0.686 + 0.727i)T \)
61 \( 1 + (-0.918 - 0.396i)T \)
67 \( 1 + (0.802 + 0.597i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (-0.597 + 0.802i)T \)
79 \( 1 + (-0.396 - 0.918i)T \)
83 \( 1 + (0.0581 - 0.998i)T \)
89 \( 1 + (-0.230 + 0.973i)T \)
97 \( 1 + (0.727 + 0.686i)T \)
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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

−18.38690379942717229430111460860, −17.995495489062867075071629143027, −16.967643292925613484955911334578, −16.05063474793172748260460248136, −15.586615062845821164217386155015, −14.996396409548732190236047287153, −14.12367869674683990506916497391, −13.93755979358772251938793436472, −12.8908807783520900817415263757, −12.81652506455023906433419795838, −11.28856166961235688680920628405, −10.83134906391262722449066728135, −10.532961933378537355715842842135, −9.44143758894077673166319597406, −8.86631987742266220770546933846, −8.13372493825041336269617864715, −7.20402182042127704349564432465, −6.12462226035254696549465765493, −5.510096996002218899477083835599, −4.92758711483820196682726059795, −4.33606750119788980902953214742, −3.23998586292837981883541361769, −2.63139792882089121256260730126, −2.07683555835671137077155570154, −1.14820158209000403782254824455, 1.233615641803492659987820782958, 2.03337116489027777344723950896, 2.36262883808129268517037544903, 3.50105259887634149350798354502, 4.15735753948555459188649430225, 5.13157752166175053593883920224, 5.78719851368964861888992202687, 6.465775049015915415771271821806, 7.38605713872823958853609705037, 7.66901041627283895617045065227, 8.572045532254886963043800658738, 9.32974335015178556446360549564, 10.24956829635714798207597993449, 11.238844064743279793207687034478, 11.59326820774315505417194932058, 12.780361488391002689506638287714, 13.18282027662949807260689062784, 13.71706168816892511923867793998, 14.20581343137055468948865077781, 14.85256386483266858400651259813, 15.52601699716006769161476007993, 16.360697768689588837024038864889, 17.21782751308979737479112931313, 17.84246752487936182898213798405, 18.254207743235359460275990468071

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