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Properties

Label 1-4033-4033.182-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.407 - 0.913i$
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  + (0.173 − 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.993 − 0.116i)5-s + (−0.993 − 0.116i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.286 + 0.957i)12-s + (0.396 + 0.918i)13-s + (−0.686 − 0.727i)14-s + (−0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (−0.0581 − 0.998i)3-s + (−0.939 − 0.342i)4-s + (−0.993 − 0.116i)5-s + (−0.993 − 0.116i)6-s + (0.597 − 0.802i)7-s + (−0.5 + 0.866i)8-s + (−0.993 + 0.116i)9-s + (−0.286 + 0.957i)10-s + (−0.286 − 0.957i)11-s + (−0.286 + 0.957i)12-s + (0.396 + 0.918i)13-s + (−0.686 − 0.727i)14-s + (−0.0581 + 0.998i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.407 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.407 - 0.913i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.407 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5227079284 - 0.3392266836i\)
\(L(\frac12)\) \(\approx\) \(0.5227079284 - 0.3392266836i\)
\(L(1)\) \(\approx\) \(0.4662188465 - 0.5974462725i\)
\(L(1)\) \(\approx\) \(0.4662188465 - 0.5974462725i\)

Euler product

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

−18.25212436834640027889786860026, −17.94142239112404067900315611045, −16.92047347003169135228360545229, −16.53794150299340060146340691561, −15.53905721457514282242194587818, −15.24711621318717868261582344594, −14.92902851605873438274124852362, −14.26780912820462651109649799518, −13.1398749769775316787589569628, −12.37650928794391892231322074043, −11.8843556217248865801883632149, −10.94895463974771798321974542456, −10.18651724733326320883521436872, −9.563582269874689085530981736290, −8.56167488914104186857300442608, −7.98362891228703426993125029525, −7.854987754493770803371205311897, −6.40708693295471229392193094002, −5.87751798139418672605797710692, −5.10266942377010265916258903899, −4.39712882152860606123011506406, −3.88163583677628004598826590197, −3.107023733646651571637664037586, −1.96469376605350283864381481402, −0.234843853279788184505130122021, 0.7774840916316317216765688507, 1.37450368717135672829586417910, 2.36292764071058196950397283579, 3.23780145911094384166786586394, 3.878171977011452336496956341748, 4.74196358879451210970380132505, 5.391486647576448882144750749599, 6.472575492783705371118072681278, 7.27420108759488278935801654835, 7.89590543820084989391358656364, 8.72028650920006107825952075357, 9.05978241881022816256755302643, 10.48742055632223819856182881285, 11.02208811053772552771257907918, 11.49794079733282110446800616611, 12.054661719473984229806704540375, 12.755955827184246161098832985225, 13.668499126369119091338670722423, 13.90995157892633681070781511642, 14.53403835729436531202442179230, 15.63845775337061327079759858919, 16.472078067508083667933936043407, 17.108722696520725260857380718085, 18.026445501693844942265368006553, 18.44792943897439867922308129558

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