Properties

Degree 1
Conductor 4003
Sign $0.620 - 0.784i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.874 + 0.484i)2-s + (−0.983 + 0.179i)3-s + (0.530 + 0.847i)4-s + (−0.619 + 0.785i)5-s + (−0.947 − 0.319i)6-s + (−0.302 − 0.953i)7-s + (0.0541 + 0.998i)8-s + (0.935 − 0.353i)9-s + (−0.922 + 0.386i)10-s + (−0.561 − 0.827i)11-s + (−0.674 − 0.738i)12-s + (−0.994 + 0.108i)13-s + (0.197 − 0.980i)14-s + (0.468 − 0.883i)15-s + (−0.436 + 0.899i)16-s + (0.267 + 0.963i)17-s + ⋯
L(s,χ)  = 1  + (0.874 + 0.484i)2-s + (−0.983 + 0.179i)3-s + (0.530 + 0.847i)4-s + (−0.619 + 0.785i)5-s + (−0.947 − 0.319i)6-s + (−0.302 − 0.953i)7-s + (0.0541 + 0.998i)8-s + (0.935 − 0.353i)9-s + (−0.922 + 0.386i)10-s + (−0.561 − 0.827i)11-s + (−0.674 − 0.738i)12-s + (−0.994 + 0.108i)13-s + (0.197 − 0.980i)14-s + (0.468 − 0.883i)15-s + (−0.436 + 0.899i)16-s + (0.267 + 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.620 - 0.784i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.620 - 0.784i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4003\)
\( \varepsilon \)  =  $0.620 - 0.784i$
motivic weight  =  \(0\)
character  :  $\chi_{4003} (1444, \cdot )$
Sato-Tate  :  $\mu(87)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 4003,\ (0:\ ),\ 0.620 - 0.784i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.4048799575 - 0.1960663065i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.4048799575 - 0.1960663065i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8185436118 + 0.3730152039i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8185436118 + 0.3730152039i\)

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

−18.625681714762937443564997029167, −18.09578034396046172289251331478, −17.1215815587083124358739921392, −16.23908124313256323299820059495, −15.916401286047718938508879573175, −15.28769136940798717666152150631, −14.51599919943214635687921824798, −13.59141355963695925743298864219, −12.632385986308598233476457675273, −12.393843572210774641586644893816, −12.023952780518317737502805346617, −11.343801235105223466091904320723, −10.4053850994195902542399739017, −9.759631203204789265193322505444, −9.15718497517259770831329337053, −7.74453907195498190428141400875, −7.37662519652042060282278106259, −6.35703285977000713608490389428, −5.58520292904200967768422994675, −5.04596918012775158424992650808, −4.60492875006042317924357995542, −3.68911198047597033339793092108, −2.57500337831909595862956700941, −1.97563708307322871737551356916, −0.84931815195266157379405183366, 0.12849824693480668810161939075, 1.53132293179331766384845007717, 2.94740677181643450747009260449, 3.40820120703949747342596211250, 4.21344409531182291831409823595, 4.89266852192835160808493692196, 5.65604139774134179870915034760, 6.43457868894396933259466363208, 7.000654604381428510900102863716, 7.58692457338608711571185258919, 8.210190855206032565834089692196, 9.630524494861688592042695748463, 10.37353593698545437376314756027, 10.988465810431395232277603963275, 11.54216727420679896045646835724, 12.150885911255113760277643412648, 13.05862108391337719304813315119, 13.47579697824384515067626270825, 14.488500974275788838688457974376, 14.91751038738574832395226275569, 15.806698428757293943310176738146, 16.27483971159401685551274333491, 16.75446482818758147214260259744, 17.64054159241543454978866827166, 18.05040954962410108758058073694

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