↑

Properties

Label 1-4033-4033.2750-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.896 + 0.443i$
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 + (−0.835 + 0.549i)3-s + 4-s + (0.597 − 0.802i)5-s + (−0.835 + 0.549i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (−0.835 + 0.549i)12-s + (0.597 − 0.802i)13-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 2-s + (−0.835 + 0.549i)3-s + 4-s + (0.597 − 0.802i)5-s + (−0.835 + 0.549i)6-s + (0.396 + 0.918i)7-s + 8-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)10-s + (0.597 + 0.802i)11-s + (−0.835 + 0.549i)12-s + (0.597 − 0.802i)13-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2750, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.896 + 0.443i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.700313372 + 0.8648270276i\)
\(L(\frac12)\) \(\approx\) \(3.700313372 + 0.8648270276i\)
\(L(1)\) \(\approx\) \(2.009105719 + 0.2731419636i\)
\(L(1)\) \(\approx\) \(2.009105719 + 0.2731419636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.835 + 0.549i)T \)
5 \( 1 + (0.597 - 0.802i)T \)
7 \( 1 + (0.396 + 0.918i)T \)
11 \( 1 + (0.597 + 0.802i)T \)
13 \( 1 + (0.597 - 0.802i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.173 - 0.984i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (0.396 + 0.918i)T \)
31 \( 1 + (-0.286 - 0.957i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.835 - 0.549i)T \)
53 \( 1 + (0.893 + 0.448i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (-0.686 - 0.727i)T \)
67 \( 1 + (-0.835 - 0.549i)T \)
71 \( 1 + T \)
73 \( 1 + (0.396 - 0.918i)T \)
79 \( 1 + (-0.0581 + 0.998i)T \)
83 \( 1 + (0.973 - 0.230i)T \)
89 \( 1 + (-0.993 - 0.116i)T \)
97 \( 1 + (-0.286 + 0.957i)T \)
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

−18.39448904228152737618894756337, −17.68602283878916465477081976812, −16.887003069411585302638792011759, −16.477685196371690313487161151655, −15.82123680536064393167521722140, −14.65703895809551054526866925968, −14.12276716002436019359868245902, −13.695192838792429756449193509689, −13.184414268572375855303429507678, −12.20176730743620434975548485039, −11.46457311168884053810488559765, −11.0802752911997748872129031872, −10.546058969453877752058012005213, −9.72635003249674955200057981266, −8.47541143465954934603349316640, −7.5124779278352273007891480679, −6.87687308404360651139866151243, −6.44920540354042725778256538092, −5.83278357156570004475094163742, −4.98597920240707390467373812587, −4.22821833152491939931708015933, −3.463974388352494173976839816283, −2.50622692567387020555644905419, −1.63176427957351824936969833535, −0.9807014865975484357029516325, 1.06786336055337074724842381372, 1.74774254061722376908968918895, 2.70969473978623468005770890446, 3.73091619741596079329946886551, 4.53491909967661082477252767190, 5.0408078519316586823776514061, 5.66193015921485903333440325666, 6.20565201683509123232323271131, 6.952391636512145679588987886333, 8.02914709366665696232240409122, 9.02392270176199424991735537057, 9.47674581027577394110262182651, 10.5231506368866804309371071849, 11.067230737726537981973505562718, 11.77532929607964638650894443600, 12.469914838202898846302694064461, 12.87262409877689828133011936341, 13.56833131778957973688095698310, 14.63775839866667747829606659440, 15.27536655060444713477072298091, 15.52227524763562063441762535912, 16.449824153446430960460013930344, 17.04169235567575118880852082770, 17.75866692825461495413773222678, 18.10889757757164889420169546080

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