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Properties

Label 1-2001-2001.905-r1-0-0
Degree $1$
Conductor $2001$
Sign $0.998 + 0.0590i$
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  + (−0.0611 + 0.998i)2-s + (−0.992 − 0.122i)4-s + (−0.996 + 0.0815i)5-s + (0.0203 − 0.999i)7-s + (0.182 − 0.983i)8-s + (−0.0203 − 0.999i)10-s + (−0.377 + 0.925i)11-s + (−0.452 − 0.891i)13-s + (0.996 + 0.0815i)14-s + (0.970 + 0.242i)16-s + (0.841 − 0.540i)17-s + (0.992 + 0.122i)19-s + (0.999 + 0.0407i)20-s + (−0.900 − 0.433i)22-s + (0.986 − 0.162i)25-s + (0.917 − 0.396i)26-s + ⋯
L(s)  = 1  + (−0.0611 + 0.998i)2-s + (−0.992 − 0.122i)4-s + (−0.996 + 0.0815i)5-s + (0.0203 − 0.999i)7-s + (0.182 − 0.983i)8-s + (−0.0203 − 0.999i)10-s + (−0.377 + 0.925i)11-s + (−0.452 − 0.891i)13-s + (0.996 + 0.0815i)14-s + (0.970 + 0.242i)16-s + (0.841 − 0.540i)17-s + (0.992 + 0.122i)19-s + (0.999 + 0.0407i)20-s + (−0.900 − 0.433i)22-s + (0.986 − 0.162i)25-s + (0.917 − 0.396i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.998 + 0.0590i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (905, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ 0.998 + 0.0590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178968604 + 0.03481872955i\)
\(L(\frac12)\) \(\approx\) \(1.178968604 + 0.03481872955i\)
\(L(1)\) \(\approx\) \(0.7399854006 + 0.2406271639i\)
\(L(1)\) \(\approx\) \(0.7399854006 + 0.2406271639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0611 + 0.998i)T \)
5 \( 1 + (-0.996 + 0.0815i)T \)
7 \( 1 + (0.0203 - 0.999i)T \)
11 \( 1 + (-0.377 + 0.925i)T \)
13 \( 1 + (-0.452 - 0.891i)T \)
17 \( 1 + (0.841 - 0.540i)T \)
19 \( 1 + (0.992 + 0.122i)T \)
31 \( 1 + (0.591 - 0.806i)T \)
37 \( 1 + (-0.685 + 0.728i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.591 + 0.806i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.794 + 0.607i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (-0.262 - 0.965i)T \)
67 \( 1 + (-0.377 - 0.925i)T \)
71 \( 1 + (0.818 - 0.574i)T \)
73 \( 1 + (0.714 - 0.699i)T \)
79 \( 1 + (-0.970 + 0.242i)T \)
83 \( 1 + (-0.557 - 0.830i)T \)
89 \( 1 + (0.882 + 0.470i)T \)
97 \( 1 + (0.301 + 0.953i)T \)
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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

−19.54354911036899924031490002684, −19.010563037293067699823187812347, −18.77582953077982558378656389063, −17.80801620438997918161331343195, −16.9188413526238564604056548415, −16.043940202373938389325508428680, −15.49542147793929058360507241247, −14.33007995307641394785562303311, −14.02175128307987417907991942846, −12.75422633328790528663586964120, −12.26053724068021222468197375935, −11.676355741320826740206331778884, −11.06517293917792910753976245843, −10.20692808713982429559112148806, −9.2414188341588652483665765735, −8.65498640431526629194896487724, −8.007268348341454012872900021109, −7.10018952109806452998394023093, −5.730955882401760145764503286206, −5.15468395315118394470193395167, −4.17596151584131635910316485594, −3.34597188733352820283028074061, −2.73431162696580024112705655735, −1.64491163931461841600491859611, −0.61014412435301117164822907412, 0.39606588592491517980936594937, 1.16169115057833351397832178310, 2.96286557290257658420170011932, 3.6837785193738707577074066877, 4.689181080742480307816134542653, 5.05576773520655274708197670742, 6.268960823811022832280211148672, 7.19769672033824473604693205258, 7.76358993884341246177558327871, 7.948462940274658597820643165569, 9.32946219413350554571369050256, 10.008872479199671064716516508508, 10.63619980762437739973004148916, 11.817079023881992179362433848884, 12.49046591215866396438114901655, 13.3155918727984866080118226449, 14.07496503447477302488082718817, 14.83317050536305746880306265467, 15.4349585091851809903168403000, 16.06234601860439536505090709739, 16.833261797705573897681352008107, 17.423275359914038354784982640746, 18.278168104099146144596234740007, 18.825179997385894658881825695492, 19.91339614405077871881980075091

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