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Properties

Label 1-1003-1003.81-r0-0-0
Degree $1$
Conductor $1003$
Sign $-0.492 - 0.870i$
Analytic cond. $4.65791$
Root an. cond. $4.65791$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.947 − 0.319i)2-s + (−0.515 − 0.856i)3-s + (0.796 − 0.605i)4-s + (0.928 − 0.370i)5-s + (−0.762 − 0.647i)6-s + (0.419 − 0.907i)7-s + (0.561 − 0.827i)8-s + (−0.468 + 0.883i)9-s + (0.762 − 0.647i)10-s + (0.963 − 0.267i)11-s + (−0.928 − 0.370i)12-s + (0.468 + 0.883i)13-s + (0.108 − 0.994i)14-s + (−0.796 − 0.605i)15-s + (0.267 − 0.963i)16-s + ⋯
L(s)  = 1  + (0.947 − 0.319i)2-s + (−0.515 − 0.856i)3-s + (0.796 − 0.605i)4-s + (0.928 − 0.370i)5-s + (−0.762 − 0.647i)6-s + (0.419 − 0.907i)7-s + (0.561 − 0.827i)8-s + (−0.468 + 0.883i)9-s + (0.762 − 0.647i)10-s + (0.963 − 0.267i)11-s + (−0.928 − 0.370i)12-s + (0.468 + 0.883i)13-s + (0.108 − 0.994i)14-s + (−0.796 − 0.605i)15-s + (0.267 − 0.963i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $-0.492 - 0.870i$
Analytic conductor: \(4.65791\)
Root analytic conductor: \(4.65791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1003} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1003,\ (0:\ ),\ -0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.544347226 - 2.649373746i\)
\(L(\frac12)\) \(\approx\) \(1.544347226 - 2.649373746i\)
\(L(1)\) \(\approx\) \(1.593447126 - 1.224343520i\)
\(L(1)\) \(\approx\) \(1.593447126 - 1.224343520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.947 - 0.319i)T \)
3 \( 1 + (-0.515 - 0.856i)T \)
5 \( 1 + (0.928 - 0.370i)T \)
7 \( 1 + (0.419 - 0.907i)T \)
11 \( 1 + (0.963 - 0.267i)T \)
13 \( 1 + (0.468 + 0.883i)T \)
19 \( 1 + (-0.976 - 0.214i)T \)
23 \( 1 + (-0.986 + 0.161i)T \)
29 \( 1 + (0.319 - 0.947i)T \)
31 \( 1 + (0.214 + 0.976i)T \)
37 \( 1 + (0.827 - 0.561i)T \)
41 \( 1 + (0.986 + 0.161i)T \)
43 \( 1 + (-0.267 + 0.963i)T \)
47 \( 1 + (-0.370 + 0.928i)T \)
53 \( 1 + (-0.647 + 0.762i)T \)
61 \( 1 + (0.319 + 0.947i)T \)
67 \( 1 + (-0.561 + 0.827i)T \)
71 \( 1 + (-0.928 - 0.370i)T \)
73 \( 1 + (0.108 - 0.994i)T \)
79 \( 1 + (-0.515 + 0.856i)T \)
83 \( 1 + (-0.0541 + 0.998i)T \)
89 \( 1 + (-0.947 - 0.319i)T \)
97 \( 1 + (-0.108 - 0.994i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.020191077849657963211153623722, −21.422985830652039650491961888204, −20.69215217851531436240315968862, −19.99995514613908254760828247204, −18.55014133006756558688104247732, −17.66040270128506103259546710311, −17.17777619949258037472066996988, −16.27988892129567423799479831455, −15.43378590333929339525740243348, −14.71988224804773978106892963150, −14.36861071597885813109344730009, −13.176660746917996817491480453, −12.35112108675379719060119869490, −11.59100958116032353064414097423, −10.80978145447551489472463630430, −10.01376848880194168181375124422, −8.98770825688463951435296859912, −8.13913513118163030026816238452, −6.65774160524725541078915320642, −6.068663348310809466951570457737, −5.49936641531632848144655396663, −4.594765791726444318159456707990, −3.68103302554665666177408064226, −2.68364588041814429838894909551, −1.69274357291071592890530182495, 1.11209589901762445613916417076, 1.61711601009480557238196386432, 2.61355869994485514318204823800, 4.148400769374775309647687825531, 4.63542180182912771127768844620, 5.97696124305161015570852371787, 6.27546641740296992167930134281, 7.12577597174091392338060083413, 8.24260474973807273564677815951, 9.43082992575310460060892521912, 10.46172531614282183182056142228, 11.19341914169304234338928644568, 11.86702786221045351703384518980, 12.77760631855107914773069170343, 13.442305622815820671518442193267, 14.11749522112822303174950771712, 14.47181909819033632485746298795, 16.11132456518556971383181549165, 16.688363862459117914192449874737, 17.42785044266619092409922993722, 18.18060210942474739420722991171, 19.42542479344198756168700259169, 19.68798829065463352453386964992, 20.85822532141377077973292408271, 21.40868548154315561234154461851

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