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Properties

Label 1-4003-4003.3784-r0-0-0
Degree $1$
Conductor $4003$
Sign $0.620 + 0.784i$
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  + (0.874 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (3784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4048799575 + 0.1960663065i\)
\(L(\frac12)\) \(\approx\) \(0.4048799575 + 0.1960663065i\)
\(L(1)\) \(\approx\) \(0.8185436118 - 0.3730152039i\)
\(L(1)\) \(\approx\) \(0.8185436118 - 0.3730152039i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (0.874 - 0.484i)T \)
3 \( 1 + (-0.983 - 0.179i)T \)
5 \( 1 + (-0.619 - 0.785i)T \)
7 \( 1 + (-0.302 + 0.953i)T \)
11 \( 1 + (-0.561 + 0.827i)T \)
13 \( 1 + (-0.994 - 0.108i)T \)
17 \( 1 + (0.267 - 0.963i)T \)
19 \( 1 + (0.700 + 0.713i)T \)
23 \( 1 + (-0.436 - 0.899i)T \)
29 \( 1 + (0.700 - 0.713i)T \)
31 \( 1 + (-0.999 - 0.0361i)T \)
37 \( 1 + (0.700 - 0.713i)T \)
41 \( 1 + (-0.561 - 0.827i)T \)
43 \( 1 + (0.336 - 0.941i)T \)
47 \( 1 + (-0.436 + 0.899i)T \)
53 \( 1 + (-0.436 + 0.899i)T \)
59 \( 1 + (-0.891 + 0.452i)T \)
61 \( 1 + (0.750 - 0.661i)T \)
67 \( 1 + (0.197 - 0.980i)T \)
71 \( 1 + (-0.983 - 0.179i)T \)
73 \( 1 + (-0.725 + 0.687i)T \)
79 \( 1 + (0.750 + 0.661i)T \)
83 \( 1 + (0.874 - 0.484i)T \)
89 \( 1 + (0.197 + 0.980i)T \)
97 \( 1 + (0.197 - 0.980i)T \)
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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

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

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