Properties

Label 1-4033-4033.1759-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.477 + 0.878i$
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  + i·2-s + (−0.396 − 0.918i)3-s − 4-s + (0.957 + 0.286i)5-s + (0.918 − 0.396i)6-s + (−0.686 − 0.727i)7-s i·8-s + (−0.686 + 0.727i)9-s + (−0.286 + 0.957i)10-s + (0.286 + 0.957i)11-s + (0.396 + 0.918i)12-s + (0.957 + 0.286i)13-s + (0.727 − 0.686i)14-s + (−0.116 − 0.993i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.396 − 0.918i)3-s − 4-s + (0.957 + 0.286i)5-s + (0.918 − 0.396i)6-s + (−0.686 − 0.727i)7-s i·8-s + (−0.686 + 0.727i)9-s + (−0.286 + 0.957i)10-s + (0.286 + 0.957i)11-s + (0.396 + 0.918i)12-s + (0.957 + 0.286i)13-s + (0.727 − 0.686i)14-s + (−0.116 − 0.993i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.028021617 + 1.206758203i\)
\(L(\frac12)\) \(\approx\) \(2.028021617 + 1.206758203i\)
\(L(1)\) \(\approx\) \(1.023567517 + 0.3165169087i\)
\(L(1)\) \(\approx\) \(1.023567517 + 0.3165169087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.396 - 0.918i)T \)
5 \( 1 + (0.957 + 0.286i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (0.286 + 0.957i)T \)
13 \( 1 + (0.957 + 0.286i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (0.727 - 0.686i)T \)
31 \( 1 + (0.549 - 0.835i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (0.597 - 0.802i)T \)
59 \( 1 + (0.116 + 0.993i)T \)
61 \( 1 + (0.998 - 0.0581i)T \)
67 \( 1 + (-0.396 + 0.918i)T \)
71 \( 1 + T \)
73 \( 1 + (0.686 - 0.727i)T \)
79 \( 1 + (0.116 + 0.993i)T \)
83 \( 1 + (0.893 + 0.448i)T \)
89 \( 1 + (0.230 + 0.973i)T \)
97 \( 1 + (0.549 + 0.835i)T \)
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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

−18.25465953251222484946087601224, −17.60840018149637561433874915413, −16.89384081925566770739828306727, −16.09899961512231651132572773861, −15.74356910800274227697338934804, −14.55972626997352826791942748285, −13.926062165051024782285483526203, −13.43626049166724551429552943162, −12.519342831497456141779893757090, −11.93923954146602421940843137967, −11.27373922206196985667702157615, −10.47855836995685153011164859777, −10.0002432166730659451481358973, −9.257068247199216233526153849132, −8.86785660971743641275718184442, −8.20810241326097017390965508533, −6.50786793292998669607865176871, −5.94282181722581878352083040274, −5.31976864345875411103055131262, −4.746039216411075591611937804761, −3.535099099361855487153244720240, −3.1913238575052176068048913474, −2.3991942594584935670445817733, −1.147617794833319542787085437229, −0.576983571140890198790555596883, 0.7022983247095825694124558622, 1.38525935350867455557766191379, 2.29936059706016213717882584571, 3.5537737418641277765886560666, 4.20534630705072980896833033730, 5.37269217743525812173809895553, 5.93952753512360407108189006356, 6.512549679923071479097893924598, 6.95274362001984014721616399856, 7.82103006278243589224555418672, 8.36582125859733422945093760758, 9.47078346758403851376170616171, 10.05980468921493956101201425579, 10.514523476338560859979480615178, 11.826902053477313657953533458374, 12.459901756283220292645778994730, 13.19486299266714536040700293714, 13.70436339235725866135855254406, 14.16704253512543428877084597664, 14.87197158339508831017003952057, 15.88375059827514554050458856797, 16.73337922003631237882114747787, 16.86646297346104327351754498542, 17.82484697571850667059807472976, 18.08454258880759121807189580791

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