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Properties

Label 1-4003-4003.1082-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.414 - 0.910i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.184 − 0.982i)2-s + (−0.413 + 0.910i)3-s + (−0.931 − 0.363i)4-s + (0.348 − 0.937i)5-s + (0.818 + 0.574i)6-s + (−0.378 − 0.925i)7-s + (−0.529 + 0.848i)8-s + (−0.658 − 0.752i)9-s + (−0.856 − 0.515i)10-s + (−0.884 + 0.466i)11-s + (0.716 − 0.697i)12-s + (−0.480 + 0.876i)13-s + (−0.979 + 0.201i)14-s + (0.709 + 0.704i)15-s + (0.735 + 0.677i)16-s + (0.675 − 0.737i)17-s + ⋯
L(s)  = 1  + (0.184 − 0.982i)2-s + (−0.413 + 0.910i)3-s + (−0.931 − 0.363i)4-s + (0.348 − 0.937i)5-s + (0.818 + 0.574i)6-s + (−0.378 − 0.925i)7-s + (−0.529 + 0.848i)8-s + (−0.658 − 0.752i)9-s + (−0.856 − 0.515i)10-s + (−0.884 + 0.466i)11-s + (0.716 − 0.697i)12-s + (−0.480 + 0.876i)13-s + (−0.979 + 0.201i)14-s + (0.709 + 0.704i)15-s + (0.735 + 0.677i)16-s + (0.675 − 0.737i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.414 - 0.910i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6625521389 - 1.029309457i\)
\(L(\frac12)\) \(\approx\) \(0.6625521389 - 1.029309457i\)
\(L(1)\) \(\approx\) \(0.7663297872 - 0.4509349565i\)
\(L(1)\) \(\approx\) \(0.7663297872 - 0.4509349565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.184 - 0.982i)T \)
3 \( 1 + (-0.413 + 0.910i)T \)
5 \( 1 + (0.348 - 0.937i)T \)
7 \( 1 + (-0.378 - 0.925i)T \)
11 \( 1 + (-0.884 + 0.466i)T \)
13 \( 1 + (-0.480 + 0.876i)T \)
17 \( 1 + (0.675 - 0.737i)T \)
19 \( 1 + (-0.0776 - 0.996i)T \)
23 \( 1 + (0.812 - 0.582i)T \)
29 \( 1 + (0.999 + 0.00941i)T \)
31 \( 1 + (0.972 + 0.233i)T \)
37 \( 1 + (0.451 + 0.892i)T \)
41 \( 1 + (0.980 - 0.196i)T \)
43 \( 1 + (0.285 + 0.958i)T \)
47 \( 1 + (-0.513 - 0.858i)T \)
53 \( 1 + (-0.404 + 0.914i)T \)
59 \( 1 + (0.258 + 0.966i)T \)
61 \( 1 + (-0.732 + 0.680i)T \)
67 \( 1 + (0.400 + 0.916i)T \)
71 \( 1 + (0.895 - 0.445i)T \)
73 \( 1 + (-0.599 - 0.800i)T \)
79 \( 1 + (0.886 - 0.462i)T \)
83 \( 1 + (-0.317 + 0.948i)T \)
89 \( 1 + (-0.521 + 0.853i)T \)
97 \( 1 + (0.632 + 0.774i)T \)
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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

−18.57493121080141692513579752180, −17.974789512763688823150203623753, −17.40578847690442393174468798515, −16.80662117888579141893974121080, −15.771900712876183678626454643606, −15.46198447663031145309188201614, −14.3741064672298461685700711754, −14.24049164978150904594372900040, −13.05602059750503625342499235723, −12.830597350681305208932328496349, −12.12251279440615939769820476826, −11.18348028221401014969513316560, −10.31429218825907765763177648971, −9.738377017621252964596309590670, −8.648789869968724691294396069668, −7.86412755725651857753042265102, −7.630809726210203935415846017579, −6.544637526683492186392209615869, −6.064832464807408752032349715387, −5.582352019850689001414039005288, −4.96354582350524220090005059220, −3.444202891766269066302447941323, −2.92445631589096861026683110929, −2.10611571301402197030256120279, −0.74599684061312815955067731588, 0.54257237950111439980933163278, 1.165620249739765270291150056099, 2.56035250526916812641848121233, 3.02305114822256838070337203879, 4.25562858641106413181399016205, 4.66722275759172391008485029235, 4.99828332915194380028743806171, 6.011489156884324587681081707388, 6.917821525699979762111125757801, 8.03440991648779745997441203005, 8.92473692126140595372996232407, 9.526609936831958250515684507847, 9.996321270884351704158973773368, 10.57882386922159619324958312122, 11.354112229417788980726056173, 12.087214699122014881866823812414, 12.639698107369909938738661805236, 13.47633411263429815002474365164, 13.92336027642548454324221930779, 14.7788874612170361349626571205, 15.622084226550131200758214072752, 16.42659523739154424819364552390, 16.86504287036866479779782438764, 17.6012029769560557880841119170, 18.11383548130523548698895226605

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