Properties

Label 1-4033-4033.2580-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.894 - 0.447i$
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 + (−0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.0581 + 0.998i)5-s + (0.993 + 0.116i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.0581 − 0.998i)12-s + (−0.893 − 0.448i)13-s + (0.286 − 0.957i)14-s + (−0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.286 + 0.957i)3-s + (−0.939 + 0.342i)4-s + (0.0581 + 0.998i)5-s + (0.993 + 0.116i)6-s + (0.893 + 0.448i)7-s + (0.5 + 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.973 − 0.230i)10-s + (0.973 + 0.230i)11-s + (−0.0581 − 0.998i)12-s + (−0.893 − 0.448i)13-s + (0.286 − 0.957i)14-s + (−0.973 − 0.230i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2580, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.894 - 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113079337 - 0.2631741855i\)
\(L(\frac12)\) \(\approx\) \(1.113079337 - 0.2631741855i\)
\(L(1)\) \(\approx\) \(0.8513756112 + 0.01456374892i\)
\(L(1)\) \(\approx\) \(0.8513756112 + 0.01456374892i\)

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 \( 1 + (-0.286 + 0.957i)T \)
5 \( 1 + (0.0581 + 0.998i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.973 + 0.230i)T \)
13 \( 1 + (-0.893 - 0.448i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.686 + 0.727i)T \)
31 \( 1 + (0.0581 - 0.998i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.396 - 0.918i)T \)
53 \( 1 + (0.893 + 0.448i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.993 + 0.116i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.973 + 0.230i)T \)
79 \( 1 + (0.0581 - 0.998i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (-0.396 - 0.918i)T \)
97 \( 1 + (-0.893 - 0.448i)T \)
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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

−18.21628712278468454619609585941, −17.55605522371812658691060956830, −17.29774990925724023247354754206, −16.59340005762676473592445755290, −16.11411520708135985650960342405, −15.07198403721117850749280628556, −14.10071260293858927115164049355, −14.02861143909153843025932477591, −13.27079989352112302097439080936, −12.264467278655922354375522897105, −11.963785626867290638132018624710, −11.07840028544296981062115181981, −9.99733220191646695057593991701, −9.26852643453299892458159170489, −8.50243653323771173391571078815, −8.03256529569292759140521974328, −7.32454576908248822234248995573, −6.66732730862824539517613446717, −5.907194991982682837350304622568, −5.180561935890786663730119384691, −4.53049559648491865502741701467, −3.88147828173657193586594934688, −2.23395488914089932138998029143, −1.40377634936888580900072948515, −0.82446559205888330450217623155, 0.473359042273017754529531940308, 1.94977336716252886493122594483, 2.44644130270770285138699494058, 3.28933202448812432495215792, 4.201348590853535689642534351727, 4.58640860421964941977000743003, 5.49758516587325559564898661808, 6.32197597799811752459628532445, 7.294107600975248890705252552460, 8.260429019752801522438749209393, 8.95372706977899740470692854992, 9.518285196982822254533696521621, 10.48817386678237665936114442291, 10.63266336695034168563537223838, 11.52711933177957284467252492962, 11.897643916412406163977416827822, 12.652685529142917008977806711752, 13.8835727957057803242707958588, 14.29363846524723178246962286235, 15.14147554574137674189093410174, 15.27289597736013839417066582618, 16.658101452517180745749924123760, 17.345731484086788481804673799502, 17.73499822500573940340676220860, 18.274730125576940961915987427773

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