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Properties

Label 1-4003-4003.117-r0-0-0
Degree $1$
Conductor $4003$
Sign $-0.0670 + 0.997i$
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  + (−0.0180 + 0.999i)2-s + (0.336 + 0.941i)3-s + (−0.999 − 0.0361i)4-s + (0.989 − 0.143i)5-s + (−0.947 + 0.319i)6-s + (−0.674 − 0.738i)7-s + (0.0541 − 0.998i)8-s + (−0.773 + 0.633i)9-s + (0.126 + 0.992i)10-s + (−0.561 + 0.827i)11-s + (−0.302 − 0.953i)12-s + (−0.994 − 0.108i)13-s + (0.750 − 0.661i)14-s + (0.468 + 0.883i)15-s + (0.997 + 0.0721i)16-s + (0.267 − 0.963i)17-s + ⋯
L(s)  = 1  + (−0.0180 + 0.999i)2-s + (0.336 + 0.941i)3-s + (−0.999 − 0.0361i)4-s + (0.989 − 0.143i)5-s + (−0.947 + 0.319i)6-s + (−0.674 − 0.738i)7-s + (0.0541 − 0.998i)8-s + (−0.773 + 0.633i)9-s + (0.126 + 0.992i)10-s + (−0.561 + 0.827i)11-s + (−0.302 − 0.953i)12-s + (−0.994 − 0.108i)13-s + (0.750 − 0.661i)14-s + (0.468 + 0.883i)15-s + (0.997 + 0.0721i)16-s + (0.267 − 0.963i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0670 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0670 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4003\)
Sign: $-0.0670 + 0.997i$
Analytic conductor: \(18.5898\)
Root analytic conductor: \(18.5898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4003} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4003,\ (0:\ ),\ -0.0670 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9374867721 + 1.002572499i\)
\(L(\frac12)\) \(\approx\) \(0.9374867721 + 1.002572499i\)
\(L(1)\) \(\approx\) \(0.7846884609 + 0.6177918431i\)
\(L(1)\) \(\approx\) \(0.7846884609 + 0.6177918431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4003 \( 1 \)
good2 \( 1 + (-0.0180 + 0.999i)T \)
3 \( 1 + (0.336 + 0.941i)T \)
5 \( 1 + (0.989 - 0.143i)T \)
7 \( 1 + (-0.674 - 0.738i)T \)
11 \( 1 + (-0.561 + 0.827i)T \)
13 \( 1 + (-0.994 - 0.108i)T \)
17 \( 1 + (0.267 - 0.963i)T \)
19 \( 1 + (-0.968 + 0.250i)T \)
23 \( 1 + (0.997 + 0.0721i)T \)
29 \( 1 + (-0.968 - 0.250i)T \)
31 \( 1 + (0.530 - 0.847i)T \)
37 \( 1 + (-0.968 - 0.250i)T \)
41 \( 1 + (-0.561 - 0.827i)T \)
43 \( 1 + (-0.983 + 0.179i)T \)
47 \( 1 + (0.997 - 0.0721i)T \)
53 \( 1 + (0.997 - 0.0721i)T \)
59 \( 1 + (0.837 + 0.546i)T \)
61 \( 1 + (0.197 + 0.980i)T \)
67 \( 1 + (0.750 + 0.661i)T \)
71 \( 1 + (0.336 + 0.941i)T \)
73 \( 1 + (-0.725 + 0.687i)T \)
79 \( 1 + (0.197 - 0.980i)T \)
83 \( 1 + (-0.0180 + 0.999i)T \)
89 \( 1 + (0.750 - 0.661i)T \)
97 \( 1 + (0.750 + 0.661i)T \)
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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

−18.66933660565212847546578011682, −17.82642494746235357770102917581, −17.10562575189806630080296929888, −16.74126726793346373669010072164, −15.20531685466620853170111690489, −14.738328576056311404539009575962, −13.93494084086099722861901919458, −13.28981566786338444190591910720, −12.81679814740643546617395029005, −12.37439213791251188324391744020, −11.531852447370367300076307342909, −10.65573079244812314038938037233, −10.09022995937843862524022604390, −9.2152004950443804395075410895, −8.73640770338519342302819860371, −8.13557465608998476225185997245, −6.96734227024516710102843413727, −6.2973996831156906794164722189, −5.51804394951950790098524477361, −4.95831301672963617297568079162, −3.43682944905590317603169357847, −3.04101552107433559157127670675, −2.19270300722105241434503881337, −1.81531657469770904427419902239, −0.63205013724675774047925448387, 0.5616696558205280178690530495, 2.11446949685647490335979577865, 2.88189681170090974174145617333, 3.89538446275307395826710911172, 4.581454111972117709537905099992, 5.27262063398079009476267334865, 5.74360000369242156851709926992, 6.974037569737745754766316100135, 7.20621238553317868782003900222, 8.25475519394782679826832944872, 9.09829801141014705439310111939, 9.5867724840382293115857919132, 10.22059329346705772823288004066, 10.47481471149497587610394443479, 11.91230298315565599605359627221, 13.00789558701705639040988892443, 13.259912382834013813862887395874, 14.06225006050283013840033587039, 14.71504765102746731738846417525, 15.21416788650276843654819923016, 15.96186251535701694451146041195, 16.73587810783453734284138312754, 17.10534020198871071261369808070, 17.528222242544686406399715615631, 18.63621597708736239439343102097

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