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Properties

Label 1-6003-6003.157-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.502 - 0.864i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.945 − 0.327i)2-s + (0.786 − 0.618i)4-s + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (0.540 − 0.841i)8-s + (0.281 − 0.959i)10-s + (−0.189 − 0.981i)11-s + (−0.235 − 0.971i)13-s + (0.0950 + 0.995i)14-s + (0.235 − 0.971i)16-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.0475 − 0.998i)20-s + (−0.5 − 0.866i)22-s + (−0.327 − 0.945i)25-s + (−0.540 − 0.841i)26-s + ⋯
L(s)  = 1  + (0.945 − 0.327i)2-s + (0.786 − 0.618i)4-s + (0.580 − 0.814i)5-s + (−0.235 + 0.971i)7-s + (0.540 − 0.841i)8-s + (0.281 − 0.959i)10-s + (−0.189 − 0.981i)11-s + (−0.235 − 0.971i)13-s + (0.0950 + 0.995i)14-s + (0.235 − 0.971i)16-s + (0.989 + 0.142i)17-s + (−0.989 + 0.142i)19-s + (−0.0475 − 0.998i)20-s + (−0.5 − 0.866i)22-s + (−0.327 − 0.945i)25-s + (−0.540 − 0.841i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.502 - 0.864i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.871473143 - 3.251729238i\)
\(L(\frac12)\) \(\approx\) \(1.871473143 - 3.251729238i\)
\(L(1)\) \(\approx\) \(1.796485930 - 0.9651862377i\)
\(L(1)\) \(\approx\) \(1.796485930 - 0.9651862377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.945 - 0.327i)T \)
5 \( 1 + (0.580 - 0.814i)T \)
7 \( 1 + (-0.235 + 0.971i)T \)
11 \( 1 + (-0.189 - 0.981i)T \)
13 \( 1 + (-0.235 - 0.971i)T \)
17 \( 1 + (0.989 + 0.142i)T \)
19 \( 1 + (-0.989 + 0.142i)T \)
31 \( 1 + (0.998 + 0.0475i)T \)
37 \( 1 + (0.909 + 0.415i)T \)
41 \( 1 + (-0.814 - 0.580i)T \)
43 \( 1 + (0.998 - 0.0475i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.235 + 0.971i)T \)
61 \( 1 + (0.458 + 0.888i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.989 + 0.142i)T \)
79 \( 1 + (-0.690 - 0.723i)T \)
83 \( 1 + (-0.580 - 0.814i)T \)
89 \( 1 + (-0.540 - 0.841i)T \)
97 \( 1 + (0.0950 - 0.995i)T \)
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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

−17.62377352737716635036901448486, −17.16363410560032310489675867478, −16.68137044516265652426049836475, −15.878596588096364098784994196, −15.091771172561199267105828017559, −14.56655246536313648319518400750, −14.04483138627226758407829002992, −13.51129249193939320524533706685, −12.78619611317272735394107984421, −12.19335570781718107642766656153, −11.33179822160294323530932268616, −10.76653505167925085923481728388, −9.99450025380512651010557299766, −9.58318250419888510336139855450, −8.31956693239646074787870878398, −7.55539090608680572131857173544, −6.91701725158899698231197267456, −6.62230209874013325480704756510, −5.781944302876839731685609184541, −4.95468070089128497764986503973, −4.1942701060367547041478087198, −3.722823463372722025360008643284, −2.666217796718432344510754971456, −2.225473665223131383481390893586, −1.25549397664132412076092323211, 0.63112130832176428874384733252, 1.413081906671219276306979821, 2.467031340230818235585354165136, 2.81659105756763594344135007541, 3.75108921252407361533657787218, 4.596586823969967863720764545677, 5.37930631733604188800981388960, 5.85681706832237910353726439962, 6.15933475191727786559538528687, 7.32999785987200152148394455878, 8.303689204740466985397697012514, 8.71161434290749521277509977268, 9.74646574642862470548703017055, 10.21705657094120881610669167890, 10.92867503181859362016889618429, 11.94739914138398024770235359925, 12.20241452915820044657969156121, 13.04355228096994013614434549942, 13.281025442324089426711311583326, 14.19057113417460990875081974102, 14.77815761723147972478410177511, 15.51665939270498874509862422124, 16.0224195990393572964096002555, 16.74344204264767085786284151619, 17.32000861213760592266191234167

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