↑

Properties

Label 1-4003-4003.208-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.992 - 0.122i$
Analytic cond. $18.5898$
Root an. cond. $18.5898$
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  + (0.958 + 0.284i)2-s + (0.700 + 0.713i)3-s + (0.837 + 0.546i)4-s + (−0.674 − 0.738i)5-s + (0.468 + 0.883i)6-s + (0.750 + 0.661i)7-s + (0.647 + 0.762i)8-s + (−0.0180 + 0.999i)9-s + (−0.436 − 0.899i)10-s + (−0.994 − 0.108i)11-s + (0.197 + 0.980i)12-s + (−0.161 + 0.986i)13-s + (0.530 + 0.847i)14-s + (0.0541 − 0.998i)15-s + (0.403 + 0.915i)16-s + (−0.370 − 0.928i)17-s + ⋯
L(s)  = 1  + (0.958 + 0.284i)2-s + (0.700 + 0.713i)3-s + (0.837 + 0.546i)4-s + (−0.674 − 0.738i)5-s + (0.468 + 0.883i)6-s + (0.750 + 0.661i)7-s + (0.647 + 0.762i)8-s + (−0.0180 + 0.999i)9-s + (−0.436 − 0.899i)10-s + (−0.994 − 0.108i)11-s + (0.197 + 0.980i)12-s + (−0.161 + 0.986i)13-s + (0.530 + 0.847i)14-s + (0.0541 − 0.998i)15-s + (0.403 + 0.915i)16-s + (−0.370 − 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.992 - 0.122i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.992 - 0.122i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1449680420 + 2.364539690i\)
\(L(\frac12)\) \(\approx\) \(-0.1449680420 + 2.364539690i\)
\(L(1)\) \(\approx\) \(1.492051558 + 1.055336481i\)
\(L(1)\) \(\approx\) \(1.492051558 + 1.055336481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.958 + 0.284i)T \)
3 \( 1 + (0.700 + 0.713i)T \)
5 \( 1 + (-0.674 - 0.738i)T \)
7 \( 1 + (0.750 + 0.661i)T \)
11 \( 1 + (-0.994 - 0.108i)T \)
13 \( 1 + (-0.161 + 0.986i)T \)
17 \( 1 + (-0.370 - 0.928i)T \)
19 \( 1 + (-0.619 + 0.785i)T \)
23 \( 1 + (0.403 + 0.915i)T \)
29 \( 1 + (-0.619 - 0.785i)T \)
31 \( 1 + (-0.891 + 0.452i)T \)
37 \( 1 + (-0.619 - 0.785i)T \)
41 \( 1 + (-0.994 + 0.108i)T \)
43 \( 1 + (-0.968 - 0.250i)T \)
47 \( 1 + (0.403 - 0.915i)T \)
53 \( 1 + (0.403 - 0.915i)T \)
59 \( 1 + (-0.983 + 0.179i)T \)
61 \( 1 + (-0.999 + 0.0361i)T \)
67 \( 1 + (0.530 - 0.847i)T \)
71 \( 1 + (0.700 + 0.713i)T \)
73 \( 1 + (0.907 + 0.419i)T \)
79 \( 1 + (-0.999 - 0.0361i)T \)
83 \( 1 + (0.958 + 0.284i)T \)
89 \( 1 + (0.530 + 0.847i)T \)
97 \( 1 + (0.530 - 0.847i)T \)
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

−18.55192203490319765800670461613, −17.55273450802423835620235731744, −16.78143044692099371946553362440, −15.53653043911962847316775212362, −15.17051217823681713097012306997, −14.74484650747046049476532234259, −14.04242443510921846063275975882, −13.22423644458143483907014141495, −12.87353808358196570607227844251, −12.13920805856055457504856951146, −11.25068466503716179973128754530, −10.593399673161847127343095533355, −10.35410665290903442015081973704, −8.91259773820688342262915251617, −7.97839598367650166719207862473, −7.638877454959222948058720712193, −6.87839761845191058737860759879, −6.29808084335169172419409546639, −5.19094654277849006230047844247, −4.47568053405150157309962122024, −3.62844149070047731909100376848, −3.02544636138156857237551223792, −2.316360966266737201015662209558, −1.54630283370188923148727144101, −0.35250136701497703203579303066, 1.84142577932060292794178140930, 2.16529038062486543626746122652, 3.33712491045275889070275282910, 3.85231209688812793537144629416, 4.81826619804914021453817324372, 5.03610587822868845378790568302, 5.76301198738298840277964008927, 7.12792579636272516785635907041, 7.64948689307149779587441679879, 8.4132289919109107544580482568, 8.84670014163653733901082970999, 9.75185611200261246211187366580, 10.85966269052396362187360763726, 11.37621820121581066283345552210, 12.0156497480600486623858599641, 12.777698600235645665485479218004, 13.553498040168476058452761667632, 14.05595284702071953265311507654, 14.92865567468319681091607119412, 15.382648325818359680684691630235, 15.82063967273793411943252908121, 16.62576618956445824307521412168, 16.988037984171816476019828919444, 18.27544145372885542084028668774, 18.96975062061577803155769129218

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