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Properties

Label 1-4033-4033.1358-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: $\chi_{4033} (1358, \cdot )$
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

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

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