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Properties

Label 1-4003-4003.1010-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: $\chi_{4003} (1010, \cdot )$
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

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

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