↑

Properties

Label 1-4033-4033.949-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.603 - 0.797i$
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 + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (0.939 + 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.5 + 0.866i)3-s + (−0.939 + 0.342i)4-s + (0.866 + 0.5i)5-s + (0.766 − 0.642i)6-s + (−0.5 + 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.5 + 0.866i)9-s + (0.342 − 0.939i)10-s + (−0.342 − 0.939i)11-s + (−0.766 − 0.642i)12-s + (0.5 + 0.866i)13-s + (0.939 + 0.342i)14-s + i·15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.603 - 0.797i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.603 - 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2013537067 - 0.4052207164i\)
\(L(\frac12)\) \(\approx\) \(0.2013537067 - 0.4052207164i\)
\(L(1)\) \(\approx\) \(0.8957813718 - 0.03618202548i\)
\(L(1)\) \(\approx\) \(0.8957813718 - 0.03618202548i\)

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 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.342 - 0.939i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.642 - 0.766i)T \)
31 \( 1 + (0.342 + 0.939i)T \)
41 \( 1 + (0.642 - 0.766i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.342 - 0.939i)T \)
53 \( 1 + (0.984 + 0.173i)T \)
59 \( 1 - T \)
61 \( 1 - iT \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.642 - 0.766i)T \)
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

−18.313339082440784446295822073949, −17.9700016823532055730088499842, −17.39104556702738031827003257050, −16.853235635406039121571802241378, −16.03895485197104804934117852611, −15.219231829474027232966988847968, −14.650044049923908496365122456487, −13.841193925799107062228896523980, −13.22128253590552244765739596051, −12.98503202652883827086968039188, −12.361001631127710154572313583065, −10.88647315501308445410259395157, −10.1756308638187720397106974221, −9.48344495214128318128762881148, −8.94913631458939268679779318374, −7.98810595549817378035284645004, −7.64935653761697596669388890872, −6.77586033822800753722928357167, −6.12475331637046849477946999663, −5.673952499580183509400579496656, −4.50604369927425494045813024364, −3.90685101206915137607367672732, −2.77310094198858929720901519001, −1.717451114725643569323400750007, −1.10023474329357468595219593421, 0.1228049196503547515775112102, 1.788045184033002344517691408994, 2.506085898755237495844933015667, 2.79864137898154963693938264205, 3.78801095874701973224044801371, 4.455832395710024631970731479263, 5.40750448399699257599594609912, 6.02181649095676291512980342034, 6.94700313609560333179260645722, 8.29510497501093847358640926644, 8.884405825394257624411031630588, 9.14522189187012009920319512155, 9.96844387330227647756881965118, 10.6741490073365136572438643219, 11.09084307043381195119391876969, 11.85085869016589265675745888334, 12.894202101932829788694174443578, 13.511992505450505210243691383790, 13.95959173229690523335677228777, 14.66468952857164035024545637481, 15.53802656259148048280805697629, 16.18104978731701742047818294560, 16.93252608562037799144193182813, 17.68269960083775289814719030846, 18.52862901809520414700059897491

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