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Properties

Label 1-2003-2003.6-r1-0-0
Degree $1$
Conductor $2003$
Sign $0.0187 - 0.999i$
Analytic cond. $215.252$
Root an. cond. $215.252$
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  + (0.970 − 0.239i)2-s + (0.675 − 0.736i)3-s + (0.885 − 0.464i)4-s + (0.539 − 0.842i)5-s + (0.479 − 0.877i)6-s + (−0.813 + 0.582i)7-s + (0.748 − 0.663i)8-s + (−0.0862 − 0.996i)9-s + (0.322 − 0.946i)10-s + (0.354 + 0.935i)11-s + (0.256 − 0.966i)12-s + (0.940 − 0.338i)13-s + (−0.650 + 0.759i)14-s + (−0.256 − 0.966i)15-s + (0.568 − 0.822i)16-s + (0.418 + 0.908i)17-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (0.675 − 0.736i)3-s + (0.885 − 0.464i)4-s + (0.539 − 0.842i)5-s + (0.479 − 0.877i)6-s + (−0.813 + 0.582i)7-s + (0.748 − 0.663i)8-s + (−0.0862 − 0.996i)9-s + (0.322 − 0.946i)10-s + (0.354 + 0.935i)11-s + (0.256 − 0.966i)12-s + (0.940 − 0.338i)13-s + (−0.650 + 0.759i)14-s + (−0.256 − 0.966i)15-s + (0.568 − 0.822i)16-s + (0.418 + 0.908i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0187 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2003 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0187 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2003\)
Sign: $0.0187 - 0.999i$
Analytic conductor: \(215.252\)
Root analytic conductor: \(215.252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2003} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2003,\ (1:\ ),\ 0.0187 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.112092404 - 5.209054278i\)
\(L(\frac12)\) \(\approx\) \(5.112092404 - 5.209054278i\)
\(L(1)\) \(\approx\) \(2.466356192 - 1.320603203i\)
\(L(1)\) \(\approx\) \(2.466356192 - 1.320603203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2003 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
3 \( 1 + (0.675 - 0.736i)T \)
5 \( 1 + (0.539 - 0.842i)T \)
7 \( 1 + (-0.813 + 0.582i)T \)
11 \( 1 + (0.354 + 0.935i)T \)
13 \( 1 + (0.940 - 0.338i)T \)
17 \( 1 + (0.418 + 0.908i)T \)
19 \( 1 + (-0.154 + 0.987i)T \)
23 \( 1 + (0.354 + 0.935i)T \)
29 \( 1 + (-0.813 + 0.582i)T \)
31 \( 1 + (0.999 + 0.0345i)T \)
37 \( 1 + (0.154 - 0.987i)T \)
41 \( 1 + (0.985 + 0.171i)T \)
43 \( 1 + (0.985 + 0.171i)T \)
47 \( 1 + (-0.479 - 0.877i)T \)
53 \( 1 + (0.962 - 0.272i)T \)
59 \( 1 + (-0.154 - 0.987i)T \)
61 \( 1 + (-0.0517 - 0.998i)T \)
67 \( 1 + (-0.120 - 0.992i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (-0.289 - 0.957i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (0.928 - 0.370i)T \)
89 \( 1 + (0.885 - 0.464i)T \)
97 \( 1 + (0.700 - 0.713i)T \)
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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

−20.21480983119805374555103283017, −19.20270850714662104459336473965, −18.89894928780573344346332017113, −17.54310678136635697997000999072, −16.63551314072102142899283304119, −16.20879181696047166670973582782, −15.48175430547577737276434154084, −14.716998722985079646469419379845, −13.98336895246356333601847290371, −13.57724982319976485585830122058, −13.10594258756048396746389478702, −11.6999916896991168774890866716, −10.98706619104040550568038115832, −10.48979828957829442899866405033, −9.53265190957734909358860066747, −8.79670186968491227492223434375, −7.7386066725206018110240941016, −6.90020116052070037518450242383, −6.256638794853275561255939134304, −5.51286732451928543221481497244, −4.33772239668017270694413605909, −3.79538707180806137114211128046, −2.7791365046728277577558737057, −2.69106396181267046854064315438, −1.04182104241639791283928351554, 0.85351569969969895695783894563, 1.70794596233461246209350682392, 2.23133085084441960055459477566, 3.45885073850000143023824837908, 3.858286227207365230118456587006, 5.136713991999260618400026179059, 6.027209519875873392849444073674, 6.31498145899057968599440077167, 7.44594170823688665604188547354, 8.24525700428799885878753991881, 9.26280060874245360587809841934, 9.724143773078122506384712134628, 10.74225989292826108016827618527, 11.99714764477959396453137399823, 12.43773214966913960237881118757, 12.95820192757643961211138444411, 13.47418818212874172647045612380, 14.35229256298766164916044607058, 15.01868256545784212716196043108, 15.726544927011304567703032026206, 16.523784677818170817252969265778, 17.42153698472711531531135837336, 18.28896141133593980914585563447, 19.13794201407732979409059778716, 19.72368218502427383005508785940

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