Properties

Label 1-4033-4033.2506-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.957 - 0.289i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s − 10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + 6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s − 10-s − 11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.957 - 0.289i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4749009245 - 0.07036723780i\)
\(L(\frac12)\) \(\approx\) \(0.4749009245 - 0.07036723780i\)
\(L(1)\) \(\approx\) \(0.5288821166 - 0.06605674394i\)
\(L(1)\) \(\approx\) \(0.5288821166 - 0.06605674394i\)

Euler product

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

−18.171788901720980382421944858278, −17.905170446032514355901362762268, −17.1517676448693265956406367823, −16.77259984415629341230008726235, −15.89500528659864588155654661673, −14.90235745789721941445727586147, −14.73918328149329164218409663577, −13.476257899073399482940279257554, −13.20182602181871346798199618213, −12.82764586225225844840491373464, −11.224028114553858100843034067729, −10.8193763949508288644574710163, −10.29256891177103762972612246272, −9.57183834464462451837811271995, −8.52760589109886578889117189379, −7.76854751676912310812682637844, −7.13582570798325991252283180149, −6.79998610685208125736529435537, −5.967497953174425353577592507406, −5.40002988035863822984324166808, −4.58850722703008231169294158053, −3.29904500893055660304103605669, −2.40923997667366997333338254406, −1.55374117322882294783388450178, −0.41394258028834268102792053323, 0.39011114531720309222363433353, 1.77570943502940509339575968591, 2.40431056262238350137465527243, 3.29585433268562216736207435931, 4.14858249091811039583321491261, 5.06030434484742553609876076380, 5.28990834099482322788211451415, 6.35114716370655534255613738775, 7.34584625095556606556853465497, 8.474537603059011152837145671630, 8.94300470277392486770359607880, 9.62903296079817488679046019388, 9.87570372306548934181606061575, 10.94298017536927440126958554834, 11.39337948300493232867933003655, 12.34453319859499954995403139893, 12.62423509016868672388286725825, 13.38996433878830614720927442609, 14.29502382057377437480206032649, 15.239754616168987211351654552419, 16.01072375889355081553220078036, 16.561222176985851646521558112156, 16.97163839784413012251177175926, 17.8009211768024933527325611334, 18.49339630173409505600311829372

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