Properties

Label 1-2001-2001.41-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.598 - 0.801i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.989 − 0.142i)8-s + (−0.755 + 0.654i)10-s + (−0.540 − 0.841i)11-s + (0.654 + 0.755i)13-s + (0.281 + 0.959i)14-s + (−0.654 + 0.755i)16-s + (−0.909 − 0.415i)17-s + (−0.909 + 0.415i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (0.989 − 0.142i)26-s + ⋯
L(s)  = 1  + (0.540 − 0.841i)2-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (−0.654 + 0.755i)7-s + (−0.989 − 0.142i)8-s + (−0.755 + 0.654i)10-s + (−0.540 − 0.841i)11-s + (0.654 + 0.755i)13-s + (0.281 + 0.959i)14-s + (−0.654 + 0.755i)16-s + (−0.909 − 0.415i)17-s + (−0.909 + 0.415i)19-s + (0.142 + 0.989i)20-s − 22-s + (0.841 + 0.540i)25-s + (0.989 − 0.142i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.598 - 0.801i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.598 - 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8699300145 - 0.4359322364i\)
\(L(\frac12)\) \(\approx\) \(0.8699300145 - 0.4359322364i\)
\(L(1)\) \(\approx\) \(0.7924255972 - 0.4005631797i\)
\(L(1)\) \(\approx\) \(0.7924255972 - 0.4005631797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.540 - 0.841i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (-0.540 - 0.841i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (-0.909 - 0.415i)T \)
19 \( 1 + (-0.909 + 0.415i)T \)
31 \( 1 + (-0.989 - 0.142i)T \)
37 \( 1 + (0.281 + 0.959i)T \)
41 \( 1 + (0.281 - 0.959i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.654 - 0.755i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (-0.989 - 0.142i)T \)
67 \( 1 + (-0.841 - 0.540i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (0.755 - 0.654i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.989 + 0.142i)T \)
97 \( 1 + (-0.281 + 0.959i)T \)
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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

−19.99400901966046885262371931989, −19.52321393254833240075299552590, −18.32261409091697308584519476268, −17.87293211229370055152471121270, −16.94930467104992988084690123207, −16.26879100653283417625041495123, −15.505871388382659127149738169635, −15.194003408227390860212338936680, −14.35800483910515391573897586360, −13.325521926040450625510179733110, −12.87154727575311718728413660606, −12.30521827328276489657720112739, −11.04328538853194508830869927741, −10.65227556855737094950692375007, −9.46092581944796015001557192990, −8.552213402372519248448177161232, −7.82495719586354893046026822571, −7.15205952350171997372030206901, −6.58287983741928150590324100538, −5.67792069879692113821096488980, −4.547144683801615164743700233268, −4.05731331662922094982534213751, −3.299089994621886344902076047099, −2.35036682377336568766366827811, −0.47399804650830954791334066687, 0.6124086855246026048706286432, 1.93890005570286136252120276874, 2.78727512586372348026696822723, 3.6366858111452486053379669088, 4.2512235333485303236157820883, 5.195993348117125724546541164798, 6.03989654438763363210667741782, 6.73077174575903825720190157325, 8.01864058448376836798917702857, 8.93673598879226424292046729782, 9.17544015758036397219135221926, 10.51220817595491636186398164522, 11.083297191153611059393772166984, 11.73542728567644634542623436677, 12.45422060944063218475391187741, 13.09599635284009897760814222335, 13.74261765450968391440573552509, 14.72687811179345881954648739706, 15.47275556030184317604573533, 15.99960867119722198340670786504, 16.70438615368598751826926988369, 18.09290512166975483734294386881, 18.67882379958172024970761524889, 19.22183433871025956786145538811, 19.72430038379204701356709943659

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