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Properties

Label 1-2001-2001.977-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.0719 + 0.997i$
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  + (0.452 + 0.891i)2-s + (−0.591 + 0.806i)4-s + (0.101 − 0.994i)5-s + (0.933 + 0.359i)7-s + (−0.986 − 0.162i)8-s + (0.933 − 0.359i)10-s + (−0.768 − 0.639i)11-s + (0.557 + 0.830i)13-s + (0.101 + 0.994i)14-s + (−0.301 − 0.953i)16-s + (−0.654 + 0.755i)17-s + (0.591 − 0.806i)19-s + (0.742 + 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.488 + 0.872i)26-s + ⋯
L(s)  = 1  + (0.452 + 0.891i)2-s + (−0.591 + 0.806i)4-s + (0.101 − 0.994i)5-s + (0.933 + 0.359i)7-s + (−0.986 − 0.162i)8-s + (0.933 − 0.359i)10-s + (−0.768 − 0.639i)11-s + (0.557 + 0.830i)13-s + (0.101 + 0.994i)14-s + (−0.301 − 0.953i)16-s + (−0.654 + 0.755i)17-s + (0.591 − 0.806i)19-s + (0.742 + 0.670i)20-s + (0.222 − 0.974i)22-s + (−0.979 − 0.202i)25-s + (−0.488 + 0.872i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0719 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0719 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.0719 + 0.997i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (977, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.0719 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321433655 + 1.420170763i\)
\(L(\frac12)\) \(\approx\) \(1.321433655 + 1.420170763i\)
\(L(1)\) \(\approx\) \(1.180948078 + 0.6012405041i\)
\(L(1)\) \(\approx\) \(1.180948078 + 0.6012405041i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.452 + 0.891i)T \)
5 \( 1 + (0.101 - 0.994i)T \)
7 \( 1 + (0.933 + 0.359i)T \)
11 \( 1 + (-0.768 - 0.639i)T \)
13 \( 1 + (0.557 + 0.830i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.591 - 0.806i)T \)
31 \( 1 + (-0.377 + 0.925i)T \)
37 \( 1 + (0.523 + 0.852i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (0.377 + 0.925i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.685 + 0.728i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.0611 + 0.998i)T \)
67 \( 1 + (0.768 - 0.639i)T \)
71 \( 1 + (-0.0203 - 0.999i)T \)
73 \( 1 + (0.182 - 0.983i)T \)
79 \( 1 + (0.301 - 0.953i)T \)
83 \( 1 + (0.339 - 0.940i)T \)
89 \( 1 + (-0.818 + 0.574i)T \)
97 \( 1 + (0.714 + 0.699i)T \)
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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

−20.115441835748122367565653927186, −18.91323587744203503870107378852, −18.30642323289903376060440931589, −17.96708803854962484461550447588, −17.19082315974771024355981499347, −15.691056607249573234366251153, −15.300011799054508291954037455523, −14.37946075165336510506780120014, −13.942155983269761301469996385284, −13.16305476183870282704617893288, −12.38135642033566910634406327374, −11.32978942048514504589969302314, −11.067980462888334695045039709302, −10.21982263818812367329330208460, −9.73663019799145103791257483365, −8.53993021737978209990857344843, −7.67972289493335363617948754919, −6.94321612346491591666099092998, −5.71555481763150506785454287989, −5.25896623137492360718053648844, −4.175266167008992940568518925201, −3.50698893044993386250235622077, −2.46338067444736884571121627600, −1.96197143474126529422085251480, −0.666129179790229321659704514959, 1.01291809494979230578622612259, 2.1443105655535856611180234004, 3.29581301441251226800058658976, 4.402331253809100752381260688143, 4.84022371873745260769863732605, 5.6423663067121089897269721922, 6.3122457834913271458087468570, 7.40309654147070573155487963343, 8.18106793933012599040032897071, 8.76625922089602004761629206679, 9.206537842265333858595428389176, 10.57381972812527001627939265400, 11.5362019547687017996199245742, 12.08025982378621509367796289450, 13.14145774362309473994470555522, 13.48608078568972921694218489763, 14.23527253803919427104577191160, 15.218287190271680401217638494485, 15.6971257878114858053894938869, 16.506268178791915713999144758308, 16.992222702516268677320877955156, 18.00885051036158619516778003715, 18.234039312035422114807420254495, 19.40674769502120592207655815730, 20.36983202454219265879098661545

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