↑

Properties

Label 1-4033-4033.645-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.184 - 0.982i$
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  − 2-s + (0.973 − 0.230i)3-s + 4-s + (−0.0581 − 0.998i)5-s + (−0.973 + 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (0.0581 + 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.973 − 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.286 − 0.957i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  − 2-s + (0.973 − 0.230i)3-s + 4-s + (−0.0581 − 0.998i)5-s + (−0.973 + 0.230i)6-s + (0.893 + 0.448i)7-s − 8-s + (0.893 − 0.448i)9-s + (0.0581 + 0.998i)10-s + (0.0581 − 0.998i)11-s + (0.973 − 0.230i)12-s + (0.0581 + 0.998i)13-s + (−0.893 − 0.448i)14-s + (−0.286 − 0.957i)15-s + 16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.184 - 0.982i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.184 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.550728031 - 1.286662867i\)
\(L(\frac12)\) \(\approx\) \(1.550728031 - 1.286662867i\)
\(L(1)\) \(\approx\) \(1.103215452 - 0.3734117324i\)
\(L(1)\) \(\approx\) \(1.103215452 - 0.3734117324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.973 - 0.230i)T \)
5 \( 1 + (-0.0581 - 0.998i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (0.0581 - 0.998i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.893 + 0.448i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.973 - 0.230i)T \)
53 \( 1 + (0.686 + 0.727i)T \)
59 \( 1 + (0.286 + 0.957i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (-0.973 - 0.230i)T \)
71 \( 1 + T \)
73 \( 1 + (0.893 - 0.448i)T \)
79 \( 1 + (0.286 + 0.957i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (-0.835 + 0.549i)T \)
97 \( 1 + (-0.993 - 0.116i)T \)
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

−18.677403630968508658735497521405, −17.78525864356835469789673955377, −17.63203046569433141048397747136, −16.69211357021785768699577016190, −15.76775059826264726155629609567, −15.05292857031542107139041874393, −14.74688098656528063830791861160, −14.296257419283480085321766139605, −13.07693402298223892108064234051, −12.48897836840765852915045645265, −11.432617483527811650879086983851, −10.696688787661514325299773736920, −10.30628406334267092617682510765, −9.75697764647023791752343359863, −8.826185646066551213004289544, −7.99413893119861613572706973046, −7.722703036334420705834302737786, −7.06574873667294229464760672564, −6.22775650865234300387033025899, −5.18306657856131808740932185507, −4.05487636506190964665508003302, −3.44814855835434916081140455318, −2.520865182291174764239200999013, −1.945789875517314426520013138027, −1.12094503945564410305062391652, 0.76147191684742991182064504382, 1.371149806328771287337363215835, 2.22378066100901369727457615572, 2.87581328472248596977519377707, 3.93044905131289154222244000713, 4.81413877739747637412140954905, 5.64899267574795740869219278653, 6.67960954165285158575517768203, 7.33389473268052754256630830312, 8.199560221966887775830496589936, 8.604627259565278448773691581479, 9.11474201469766810383488048962, 9.56855728878742960643949632703, 10.80366264732814348896439940865, 11.31835195953492855360857185335, 12.16025177662059576711422260256, 12.66391906468121021916612671223, 13.702697981154083528104624813737, 14.24775689852673265728494720273, 15.03502812046710531902623567090, 15.68401731497779254429996926495, 16.49115487428717449156989188262, 16.78641873021354319195265575446, 17.89485248951395238821797724940, 18.28626834349977469995089769413

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