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Properties

Label 1-2001-2001.482-r1-0-0
Degree $1$
Conductor $2001$
Sign $0.272 + 0.962i$
Analytic cond. $215.037$
Root an. cond. $215.037$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (−0.974 − 0.222i)10-s + (0.433 − 0.900i)11-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s + i·17-s + (0.974 + 0.222i)19-s + (−0.900 + 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.222 + 0.974i)25-s + (0.974 − 0.222i)26-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (−0.433 − 0.900i)8-s + (−0.974 − 0.222i)10-s + (0.433 − 0.900i)11-s + (0.900 + 0.433i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s + i·17-s + (0.974 + 0.222i)19-s + (−0.900 + 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.222 + 0.974i)25-s + (0.974 − 0.222i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.272 + 0.962i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ 0.272 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7188469342 + 0.5433320318i\)
\(L(\frac12)\) \(\approx\) \(0.7188469342 + 0.5433320318i\)
\(L(1)\) \(\approx\) \(1.232661366 - 0.5348385892i\)
\(L(1)\) \(\approx\) \(1.232661366 - 0.5348385892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.781 - 0.623i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 + iT \)
19 \( 1 + (0.974 + 0.222i)T \)
31 \( 1 + (-0.781 + 0.623i)T \)
37 \( 1 + (-0.433 - 0.900i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (0.433 - 0.900i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.974 + 0.222i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (0.433 + 0.900i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.781 + 0.623i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−20.066460252021584140203478019701, −18.73269724332085653612509157908, −17.9808905970835335214396097939, −17.44669519339748888286713146043, −16.47368596142293743454021172962, −15.878554255103781219900010963686, −15.17392941693971494995285273321, −14.52081278096978064771909349891, −13.80660196109826434275884185072, −13.29714678127087452612779316294, −12.2194565792124626442584224756, −11.5671067600433030548541501327, −10.95549919461276123911986954074, −10.04439075970997379670980443402, −9.001570901627982934944553848840, −7.91196262001809643057520900515, −7.37070001079220362464364636971, −6.89760469175600899276464048354, −6.011794657993105587411466129744, −4.9394675784540519948587033658, −4.24066010955569831424355835310, −3.49101767255687158066294385036, −2.83758004541130133274188415693, −1.49142670654890358280772944992, −0.11375964312404262570014391647, 1.207035719051994378080075250916, 1.68919975047946140224443534940, 3.03956065519470171056627659898, 3.69973128052398762014208366625, 4.42535402051752878619782243277, 5.5247917467052328480748768356, 5.79332664909071959401623369301, 6.86811450795365811197121859189, 8.09414034267599290468583666694, 8.858489864831557815333358622679, 9.29912356315406814499964782597, 10.592689063346272679050868908646, 11.23806904828483704781949860096, 11.96744187764482147300249925750, 12.33329179326226653518896305318, 13.29360671414794548450131323994, 13.87913344212607799029694323543, 14.81046973199095741910841545130, 15.42517408548636479517838783665, 16.17605944753964641480647470653, 16.69607527552209976771898035847, 18.0695682990349894490732148733, 18.63766478160826813786788906549, 19.39939543963537885247999671030, 19.915945588302161353301223203813

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