Properties

Label 1-2001-2001.8-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.999 + 0.00710i$
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  + (0.574 + 0.818i)2-s + (−0.339 + 0.940i)4-s + (0.685 + 0.728i)5-s + (−0.979 + 0.202i)7-s + (−0.965 + 0.262i)8-s + (−0.202 + 0.979i)10-s + (−0.670 − 0.742i)11-s + (−0.0203 − 0.999i)13-s + (−0.728 − 0.685i)14-s + (−0.768 − 0.639i)16-s + (−0.540 − 0.841i)17-s + (0.940 + 0.339i)19-s + (−0.917 + 0.396i)20-s + (0.222 − 0.974i)22-s + (−0.0611 + 0.998i)25-s + (0.806 − 0.591i)26-s + ⋯
L(s)  = 1  + (0.574 + 0.818i)2-s + (−0.339 + 0.940i)4-s + (0.685 + 0.728i)5-s + (−0.979 + 0.202i)7-s + (−0.965 + 0.262i)8-s + (−0.202 + 0.979i)10-s + (−0.670 − 0.742i)11-s + (−0.0203 − 0.999i)13-s + (−0.728 − 0.685i)14-s + (−0.768 − 0.639i)16-s + (−0.540 − 0.841i)17-s + (0.940 + 0.339i)19-s + (−0.917 + 0.396i)20-s + (0.222 − 0.974i)22-s + (−0.0611 + 0.998i)25-s + (0.806 − 0.591i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.999 + 0.00710i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.999 + 0.00710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336907661 + 0.004750526672i\)
\(L(\frac12)\) \(\approx\) \(1.336907661 + 0.004750526672i\)
\(L(1)\) \(\approx\) \(1.061054184 + 0.4875411227i\)
\(L(1)\) \(\approx\) \(1.061054184 + 0.4875411227i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.574 + 0.818i)T \)
5 \( 1 + (0.685 + 0.728i)T \)
7 \( 1 + (-0.979 + 0.202i)T \)
11 \( 1 + (-0.670 - 0.742i)T \)
13 \( 1 + (-0.0203 - 0.999i)T \)
17 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.940 + 0.339i)T \)
31 \( 1 + (-0.0407 - 0.999i)T \)
37 \( 1 + (0.953 + 0.301i)T \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (0.0407 - 0.999i)T \)
47 \( 1 + (-0.433 - 0.900i)T \)
53 \( 1 + (-0.970 - 0.242i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.470 + 0.882i)T \)
67 \( 1 + (-0.742 - 0.670i)T \)
71 \( 1 + (0.986 - 0.162i)T \)
73 \( 1 + (0.994 - 0.101i)T \)
79 \( 1 + (-0.639 - 0.768i)T \)
83 \( 1 + (0.933 - 0.359i)T \)
89 \( 1 + (0.983 - 0.182i)T \)
97 \( 1 + (-0.0815 - 0.996i)T \)
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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

−20.05708640943190584022197885533, −19.49133652875343687037496126208, −18.61179117046799310092008518589, −17.882995334161002380593988499854, −17.13919020250782162409476914243, −16.12772872488724018538674228036, −15.66508812633217359720115710021, −14.5706056876027399968759151665, −13.86572068908279021111324981290, −13.12902465362467613165694014900, −12.74813803157267251705962526689, −12.042808620471245018069047703898, −11.060716476008917577043324736637, −10.26742614353148809665581686782, −9.481320993854344784527427463611, −9.26189186163560216292367737548, −8.05535082309254509860043285231, −6.72126161182540365943310921353, −6.25201967027479839195579866996, −5.183435061675190982273183687906, −4.64103925051669406874812261548, −3.74272843939600032129927133958, −2.75929616791046688274556243013, −1.9553177259186430176439456295, −1.103459716401046119726823637775, 0.38745219007521809074388251483, 2.334718071091975808117169472763, 3.07320142451667258228502787105, 3.513561575767638353398805821212, 4.91995072332295049349000931779, 5.69371972453836255206560626694, 6.11655862785450385305807846255, 7.025306776850774335064792253230, 7.66206712883383492341463302738, 8.61132533551761712651195812344, 9.5160169896308144656280991482, 10.12868183319580720952763760621, 11.12945358186443936293470612069, 11.98681187897464935491895308096, 12.959464920045060671177369454470, 13.51628583188673868597571230205, 13.87947379420081302889648530592, 15.04638478050065187210497793599, 15.42632820124817280580339322852, 16.26698326876645258975676355813, 16.81934550892272891786301218364, 17.85267016573089995520154691164, 18.32224682906370647672188907932, 18.91352450500068127131543389632, 20.14958028238778614291490222706

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