↑

Properties

Label 1-4033-4033.3600-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

Related objects

Downloads

Learn more

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: $\chi_{4033} (3600, \cdot )$
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 \( 1 + (-0.893 + 0.448i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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