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Properties

Label 1-4033-4033.787-r1-0-0
Degree $1$
Conductor $4033$
Sign $0.531 + 0.847i$
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  + (−0.642 + 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (0.918 + 0.396i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (−0.448 − 0.893i)13-s + (−0.230 + 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)2-s + (−0.286 − 0.957i)3-s + (−0.173 − 0.984i)4-s + (−0.0581 + 0.998i)5-s + (0.918 + 0.396i)6-s + (0.893 − 0.448i)7-s + (0.866 + 0.5i)8-s + (−0.835 + 0.549i)9-s + (−0.727 − 0.686i)10-s + (−0.727 + 0.686i)11-s + (−0.893 + 0.448i)12-s + (−0.448 − 0.893i)13-s + (−0.230 + 0.973i)14-s + (0.973 − 0.230i)15-s + (−0.939 + 0.342i)16-s + (−0.342 − 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.531 + 0.847i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ 0.531 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043160380 + 0.5772532710i\)
\(L(\frac12)\) \(\approx\) \(1.043160380 + 0.5772532710i\)
\(L(1)\) \(\approx\) \(0.7005204341 + 0.1213802224i\)
\(L(1)\) \(\approx\) \(0.7005204341 + 0.1213802224i\)

Euler product

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

−18.05022413649804366142368134312, −17.36312326436725807069489725194, −17.05126489679000063580792612637, −16.25157598997348765222875206058, −15.69930736668213276318090273477, −15.01300503351486580479855909698, −13.9317505268707929341706704063, −13.36266502254131045019135357452, −12.29838970630197747586040982288, −11.95170935798692379318799254534, −11.1763807349279117537417827237, −10.68933190974193160551239014104, −9.92898298117311003023531423250, −8.97319302742043405234983132299, −8.78118170210665870413437096421, −8.17013736929656272675384807054, −7.19287711203873460670929518694, −5.96570990605826456166211765374, −5.18578960374822463348088553114, −4.42505939935627128103239642329, −4.16066594918126849088062924443, −2.84815276214898239534090306342, −2.28076935397366222582820024563, −1.145816762852858167235737115890, −0.39581676620432752322771242429, 0.58101874486947753265856312313, 1.42129955697447464325303535330, 2.28874755974869613273609590660, 3.01044988410869316869136508125, 4.46508601773564212691285057033, 5.28147063819407484728230038117, 5.75196993225853034178286101805, 6.84779884327348928621255315111, 7.21335702555902373703678687438, 7.829704930164266986099138700322, 8.17287036107536069987212383278, 9.40116873702287374824900499868, 10.21323422048519522395366997311, 10.744719163228328375448579379947, 11.37866675234100114014041649822, 12.12870446721235543710855827293, 13.18686735884643809245416741950, 13.76398080964992655545643419001, 14.46042800272995732248443544873, 14.93327846635645962204538043126, 15.68174280976563931175363858599, 16.493843808528768916382873696555, 17.45454219291330979267548633196, 17.76647021419050730222334641306, 18.13139380671597734670924610970

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