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Properties

Label 1-4033-4033.2962-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.999 + 0.00625i$
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  + (−0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.866 + 0.5i)5-s + (−0.939 + 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (0.939 + 0.342i)12-s + (0.5 − 0.866i)13-s + (−0.173 + 0.984i)14-s + i·15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.766 − 0.642i)2-s + (0.5 − 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.866 + 0.5i)5-s + (−0.939 + 0.342i)6-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.984 + 0.173i)10-s + (−0.984 + 0.173i)11-s + (0.939 + 0.342i)12-s + (0.5 − 0.866i)13-s + (−0.173 + 0.984i)14-s + i·15-s + (−0.939 + 0.342i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.999 + 0.00625i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.999 + 0.00625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4439664532 + 0.001388697453i\)
\(L(\frac12)\) \(\approx\) \(0.4439664532 + 0.001388697453i\)
\(L(1)\) \(\approx\) \(0.5143000023 - 0.3138186457i\)
\(L(1)\) \(\approx\) \(0.5143000023 - 0.3138186457i\)

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 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.984 + 0.173i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.342 + 0.939i)T \)
31 \( 1 + (0.984 - 0.173i)T \)
41 \( 1 + (-0.342 + 0.939i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.984 + 0.173i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 - T \)
61 \( 1 - iT \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (0.342 - 0.939i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−18.57778553591397057886337902461, −17.81344750238784478790685954386, −16.74742319072140473552146578404, −16.25343096155998305645342230646, −15.89019056983841507083256625754, −15.24961427161825571807537297603, −14.84972631846938592317265096500, −13.80055725389160488918695615025, −13.24631109593924769061421108012, −12.05411575158786948572781507327, −11.5291355728363071525996448755, −10.61769354427096836295197383652, −10.03801821466550260889992102685, −9.29694437724386252159692591345, −8.587126650961845539633379112055, −8.29165911765769556308399082675, −7.596505002942960916186434560153, −6.513057863645522554404347160823, −5.787927767858144019153714274535, −4.99349366599728524960849966633, −4.39246320447963869071152546873, −3.390662402078162953830378762845, −2.58063670106876677680764866454, −1.63308436905995533239775603106, −0.20987114502677679213276564816, 0.75712289642110901186524732362, 1.48176162960861462826927397980, 2.83053875265443099047451045129, 3.074846157558381187434528410030, 3.68394406370639133083651294657, 4.792303604244143345807924229281, 6.064865791125013443199802784424, 7.03389484323114810513954107252, 7.42137895661684338984081646324, 7.96354844294635014521694461429, 8.50847627395411608953370301803, 9.633772680886235758535734231920, 10.13632618781803260109815896445, 10.86784939302208562740382035945, 11.6760892882113627129928654201, 12.13067777752102707645532039720, 13.03141173365087482377597194539, 13.4632590538530437555366096028, 14.139916583216614244138120231067, 15.22817707705092533275531880631, 15.918234635733363132498482524021, 16.35344630287571156251901395730, 17.50474020328511297161437841334, 18.01035443392712857445439037645, 18.56078742146313150683622790118

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