Properties

Degree 1
Conductor 4003
Sign $0.625 - 0.780i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $0.625 - 0.780i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (316, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4003,\ (0:\ ),\ 0.625 - 0.780i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08154511175 - 0.03914062969i$
$L(\frac12,\chi)$  $\approx$  $0.08154511175 - 0.03914062969i$
$L(\chi,1)$  $\approx$  0.4669962063 + 0.3303830401i
$L(1,\chi)$  $\approx$  0.4669962063 + 0.3303830401i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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