↑

Properties

Label 1-4033-4033.2148-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.906 + 0.423i$
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  + (0.939 − 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.342 + 0.939i)5-s + (0.766 − 0.642i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (0.5 − 0.866i)12-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.342 + 0.939i)5-s + (0.766 − 0.642i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (0.766 − 0.642i)9-s + (0.642 + 0.766i)10-s + (−0.642 + 0.766i)11-s + (0.5 − 0.866i)12-s + (−0.766 − 0.642i)13-s + (−0.766 + 0.642i)14-s + (0.642 + 0.766i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.906 + 0.423i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.906 + 0.423i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.101415423 + 0.9103778130i\)
\(L(\frac12)\) \(\approx\) \(4.101415423 + 0.9103778130i\)
\(L(1)\) \(\approx\) \(2.345656705 - 0.1212747103i\)
\(L(1)\) \(\approx\) \(2.345656705 - 0.1212747103i\)

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

−18.77986132547333877154639303130, −17.26708292375466448252377673537, −16.786565837905489262939581625650, −16.32739920651736626252032768218, −15.56161439402816769304957869178, −15.14939489674412425883468557594, −13.99224257542175139314792862879, −13.811288198620865637151912306509, −13.04527632709022189855311686025, −12.68037807877077264153875705717, −11.792632701372685737216064351995, −10.82903606228725578906669383355, −10.03563386050160163913477832406, −9.265871706848207141260679791780, −8.75938708471916458776258829167, −7.77066898476870146679251099644, −7.32502086316768817454228194281, −6.35999585849180127315024986336, −5.57033839950006498883528009877, −4.79700203460674814168105976437, −4.2317564659182329083766432058, −3.42090491562890915796341215742, −2.63860761650183997719835595701, −2.11241134475855437954221326795, −0.68185498068537445752431503675, 1.23831402018706054640054074430, 2.222836188846342265151851221688, 2.69275283539214276107730001537, 3.28300082525412753333472014416, 3.93655468890008458774706002626, 5.06624968882876181179605447318, 5.819756331455805759731200391332, 6.6050630263932637973364110886, 7.186043652711056419095762896894, 7.75386065080516461098913229418, 8.91361192249366841513936874411, 9.799186932688258310161983275412, 10.28221243203271403965241615998, 10.709271998440603471881902807380, 12.10201985173707480299561447439, 12.531121737418237589991937970917, 13.00302234939354915607411839192, 13.69944262175166826144347922952, 14.51856917314589428919631776694, 15.02501573104780189644540597589, 15.25679310443988776515680189853, 16.21057668484728022695955584175, 17.164958496800735015471808148258, 18.17989366690026228213366108099, 18.72729663369232088346013804239

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