Properties

Degree 1
Conductor 4003
Sign $-0.992 - 0.122i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $-0.992 - 0.122i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (208, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4003,\ (0:\ ),\ -0.992 - 0.122i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(-0.1449680420 + 2.364539690i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(-0.1449680420 + 2.364539690i\)
\(L(\chi,1)\)  \(\approx\)  \(1.492051558 + 1.055336481i\)
\(L(1,\chi)\)  \(\approx\)  \(1.492051558 + 1.055336481i\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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