Properties

Label 1-4033-4033.2006-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.686 + 0.726i$
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  + (0.939 + 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (0.893 + 0.448i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (0.686 + 0.727i)12-s + (0.686 + 0.727i)13-s + (−0.396 − 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.993 + 0.116i)3-s + (0.766 + 0.642i)4-s + (−0.957 − 0.286i)5-s + (0.893 + 0.448i)6-s + (−0.686 − 0.727i)7-s + (0.5 + 0.866i)8-s + (0.973 + 0.230i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (0.686 + 0.727i)12-s + (0.686 + 0.727i)13-s + (−0.396 − 0.918i)14-s + (−0.918 − 0.396i)15-s + (0.173 + 0.984i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.686 + 0.726i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2006, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.686 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.923642277 + 1.691111950i\)
\(L(\frac12)\) \(\approx\) \(3.923642277 + 1.691111950i\)
\(L(1)\) \(\approx\) \(2.244994660 + 0.5470963916i\)
\(L(1)\) \(\approx\) \(2.244994660 + 0.5470963916i\)

Euler product

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

−18.614963268033380321032801200256, −18.17754926003263106413612827126, −16.63539865213825282193432172509, −15.90114493910537914537063917305, −15.55241933471824252755160706548, −14.89506115679152888692875806945, −14.29519207019398868453676666009, −13.74156550691659870673069178479, −12.70635550629865869620331200452, −12.36948359855757742093008855242, −11.86805394022468514865903855136, −10.860791013686587990878993244373, −10.15320544654713818262048191064, −9.39786797815605351769607461220, −8.68733568803349930999138101859, −7.71797491647262997943526902557, −7.20968607862363641339638051510, −6.36792984813170365129151720209, −5.70568407810165246309458138204, −4.54605439164047839927957615710, −3.87559780972772527398626134627, −3.383985990108225632252980076444, −2.67843722309830777453006968496, −1.93647836992658267914228666637, −0.848178532852246952480408987642, 1.04379839150396166483320134957, 2.00457872630410943968062171960, 3.16758794298844265725123193502, 3.616469105561347717453269289429, 4.15810279741520474360252097237, 4.66370399117500851798050512179, 6.076339223728695278965216945726, 6.57642017834334187829733173891, 7.38178517070158368920778164098, 7.96664328395634626715897382845, 8.712371535694994460727792493651, 9.28393248542134713429960289874, 10.45845195651779046224739101814, 11.08045532338319254007714263903, 11.89745885835375510151312917962, 12.565867109260861164724043885383, 13.32037134772513539372820330073, 13.786794589178610570844289558715, 14.37751407338403257066580827958, 15.186441302545120091438848686811, 15.717685425367733133650520788297, 16.34091284845667750393391992848, 16.7480538162575065459977661887, 17.72734708211760940457356498982, 19.01851966748491777499060437683

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