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Properties

Label 1-4003-4003.316-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.625 - 0.780i$
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  + (−0.968 + 0.250i)2-s + (−0.0901 + 0.995i)3-s + (0.874 − 0.484i)4-s + (−0.436 + 0.899i)5-s + (−0.161 − 0.986i)6-s + (0.590 + 0.806i)7-s + (−0.725 + 0.687i)8-s + (−0.983 − 0.179i)9-s + (0.197 − 0.980i)10-s + (0.468 + 0.883i)11-s + (0.403 + 0.915i)12-s + (0.0541 − 0.998i)13-s + (−0.773 − 0.633i)14-s + (−0.856 − 0.515i)15-s + (0.530 − 0.847i)16-s + (0.796 − 0.605i)17-s + ⋯
L(s)  = 1  + (−0.968 + 0.250i)2-s + (−0.0901 + 0.995i)3-s + (0.874 − 0.484i)4-s + (−0.436 + 0.899i)5-s + (−0.161 − 0.986i)6-s + (0.590 + 0.806i)7-s + (−0.725 + 0.687i)8-s + (−0.983 − 0.179i)9-s + (0.197 − 0.980i)10-s + (0.468 + 0.883i)11-s + (0.403 + 0.915i)12-s + (0.0541 − 0.998i)13-s + (−0.773 − 0.633i)14-s + (−0.856 − 0.515i)15-s + (0.530 − 0.847i)16-s + (0.796 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.625 - 0.780i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.625 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08154511175 - 0.03914062969i\)
\(L(\frac12)\) \(\approx\) \(0.08154511175 - 0.03914062969i\)
\(L(1)\) \(\approx\) \(0.4669962063 + 0.3303830401i\)
\(L(1)\) \(\approx\) \(0.4669962063 + 0.3303830401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.968 + 0.250i)T \)
3 \( 1 + (-0.0901 + 0.995i)T \)
5 \( 1 + (-0.436 + 0.899i)T \)
7 \( 1 + (0.590 + 0.806i)T \)
11 \( 1 + (0.468 + 0.883i)T \)
13 \( 1 + (0.0541 - 0.998i)T \)
17 \( 1 + (0.796 - 0.605i)T \)
19 \( 1 + (-0.922 - 0.386i)T \)
23 \( 1 + (0.530 - 0.847i)T \)
29 \( 1 + (-0.922 + 0.386i)T \)
31 \( 1 + (-0.0180 + 0.999i)T \)
37 \( 1 + (-0.922 + 0.386i)T \)
41 \( 1 + (0.468 - 0.883i)T \)
43 \( 1 + (-0.817 + 0.576i)T \)
47 \( 1 + (0.530 + 0.847i)T \)
53 \( 1 + (0.530 + 0.847i)T \)
59 \( 1 + (-0.232 - 0.972i)T \)
61 \( 1 + (0.935 - 0.353i)T \)
67 \( 1 + (-0.773 + 0.633i)T \)
71 \( 1 + (-0.0901 + 0.995i)T \)
73 \( 1 + (-0.370 - 0.928i)T \)
79 \( 1 + (0.935 + 0.353i)T \)
83 \( 1 + (-0.968 + 0.250i)T \)
89 \( 1 + (-0.773 - 0.633i)T \)
97 \( 1 + (-0.773 + 0.633i)T \)
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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

−18.83542057252858165406371207051, −17.86439631008617125314165282170, −17.05674545236354307897292270381, −16.774286809198164218344192774679, −16.46807635185111373200211011160, −15.18427405342307764859365240211, −14.5215621726787505250416089814, −13.51640105150101177190158325997, −13.10866559836725961416232100433, −12.14911508697301959819243087099, −11.64311906457336798237139801994, −11.20555916295329964793880752826, −10.385737204919017970421994049297, −9.32627642372864888363602802071, −8.70669804682827439626688321236, −8.14188332308304462026939376727, −7.58223216266531022624850384131, −6.9090993806095195231062133467, −6.078201884663033936928059295156, −5.3204311427047812790443530252, −3.949253131574624073484679653464, −3.63223450745207310080320322136, −2.21042700132081065361322654364, −1.480956741861614706630236312115, −1.02996608677166846736072813593, 0.039322935253437868840785743598, 1.49176745419588273634661275113, 2.62277077034580272731942243262, 2.98002582692602274480248874098, 4.09691464894705622849707279435, 5.07999883250812082172641934735, 5.6296782308787810298816995286, 6.57565633529318451243013032562, 7.22821780807079426878863093506, 8.095370353014480361285623703081, 8.67340218956379020583512071111, 9.38084532054183087266096057988, 10.11607292652988116843093261539, 10.73720142401686445047553084836, 11.15212764267542075948294636529, 12.00822048235216924446843660938, 12.497294869905241535613847529934, 14.18333304315004671170006838365, 14.6648286751547471652690856076, 15.13548494818862267971256648241, 15.57278638267820994909499053, 16.28806403663058461908806385132, 17.1873197139256315761669414708, 17.6467463486631488646171317775, 18.28973902351572051400920119500

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