↑

Properties

Label 1-4033-4033.326-r0-0-0
Degree $1$
Conductor $4033$
Sign $0.942 + 0.334i$
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  + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s + 6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)14-s + (0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s + 6-s + (−0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (0.173 − 0.984i)13-s + (−0.5 + 0.866i)14-s + (0.939 + 0.342i)15-s + (−0.939 − 0.342i)16-s + (0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ 0.942 + 0.334i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.919585631 + 0.6744084041i\)
\(L(\frac12)\) \(\approx\) \(3.919585631 + 0.6744084041i\)
\(L(1)\) \(\approx\) \(2.194656025 - 0.1688003905i\)
\(L(1)\) \(\approx\) \(2.194656025 - 0.1688003905i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.173 - 0.984i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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

−18.62500300633498132629676527889, −17.6637106427282697788394965351, −16.97294199134341867489832718156, −16.282434994430559028613694115118, −15.75642371967855141854534378176, −14.738431402421709298222148958823, −14.08200941249126458807192792820, −13.73079464644302699308086150429, −13.403632969941519298410109591187, −12.48942456640439025579403682608, −11.8509602420962469684896012558, −11.04014121078932571831973119810, −9.87871079882215570912087638116, −9.17258218914027648427791881057, −8.79963976205665404171690363467, −7.6850072236127535150297118770, −7.00245689167537774589234071460, −6.48518982338756715901175709718, −6.05503986566431817226022890654, −5.06429364571268375998771172614, −3.976076002229074570182607320873, −3.34637818150998503774819763201, −2.68746365828581364478795911042, −1.98729462998518156879925570920, −0.71186937665225441449767276288, 1.29151714023062299218172141971, 1.90243686682061401387756507132, 2.752469014393827247701077830536, 3.48373322060643706928986966338, 3.96661330024166446112161977151, 5.11185673677536986780240687845, 5.50655006805023082862426483466, 6.27709103347761257153556712794, 7.17173524778314141097025603125, 8.28507094833330145533439099783, 9.13059295413199727371019367446, 9.70927897611682330424092334957, 10.12324476499275029033543077781, 10.656317180790615286439591595510, 11.83854601234645707429914819394, 12.635670590457711955361562519593, 12.96759787211839455513739976031, 13.7218424660251260816052999759, 14.31081917558185621970548768094, 15.06736969166560257505064484455, 15.48175550899098343214577767601, 16.33997506011290036608706163969, 16.96065231637531904128883678645, 18.12012457515339187142786459661, 18.51518929430875009518153387807

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