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Properties

Label 1-2001-2001.458-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.981 - 0.192i$
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  + (−0.794 + 0.607i)2-s + (0.262 − 0.965i)4-s + (0.339 − 0.940i)5-s + (0.301 − 0.953i)7-s + (0.377 + 0.925i)8-s + (0.301 + 0.953i)10-s + (−0.452 − 0.891i)11-s + (0.685 + 0.728i)13-s + (0.339 + 0.940i)14-s + (−0.862 − 0.505i)16-s + (−0.654 − 0.755i)17-s + (−0.262 + 0.965i)19-s + (−0.818 − 0.574i)20-s + (0.900 + 0.433i)22-s + (−0.768 − 0.639i)25-s + (−0.986 − 0.162i)26-s + ⋯
L(s)  = 1  + (−0.794 + 0.607i)2-s + (0.262 − 0.965i)4-s + (0.339 − 0.940i)5-s + (0.301 − 0.953i)7-s + (0.377 + 0.925i)8-s + (0.301 + 0.953i)10-s + (−0.452 − 0.891i)11-s + (0.685 + 0.728i)13-s + (0.339 + 0.940i)14-s + (−0.862 − 0.505i)16-s + (−0.654 − 0.755i)17-s + (−0.262 + 0.965i)19-s + (−0.818 − 0.574i)20-s + (0.900 + 0.433i)22-s + (−0.768 − 0.639i)25-s + (−0.986 − 0.162i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.981 - 0.192i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03758270607 - 0.3864303304i\)
\(L(\frac12)\) \(\approx\) \(0.03758270607 - 0.3864303304i\)
\(L(1)\) \(\approx\) \(0.6486018106 - 0.1064487390i\)
\(L(1)\) \(\approx\) \(0.6486018106 - 0.1064487390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.794 + 0.607i)T \)
5 \( 1 + (0.339 - 0.940i)T \)
7 \( 1 + (0.301 - 0.953i)T \)
11 \( 1 + (-0.452 - 0.891i)T \)
13 \( 1 + (0.685 + 0.728i)T \)
17 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (-0.262 + 0.965i)T \)
31 \( 1 + (-0.0611 + 0.998i)T \)
37 \( 1 + (-0.947 - 0.320i)T \)
41 \( 1 + (-0.841 - 0.540i)T \)
43 \( 1 + (0.0611 + 0.998i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.933 + 0.359i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.742 - 0.670i)T \)
67 \( 1 + (0.452 - 0.891i)T \)
71 \( 1 + (-0.970 - 0.242i)T \)
73 \( 1 + (-0.591 - 0.806i)T \)
79 \( 1 + (0.862 - 0.505i)T \)
83 \( 1 + (-0.523 + 0.852i)T \)
89 \( 1 + (0.488 + 0.872i)T \)
97 \( 1 + (0.992 + 0.122i)T \)
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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

−20.27377510345502840858872088879, −19.438674139890361689669366049559, −18.69413957464570780314237910890, −18.16936654314498918239995619701, −17.632722106849938748234233181318, −17.07766437607146672872825901240, −15.61928982004517732578339358379, −15.44954523795689758369401870445, −14.64880620939584312988897120712, −13.27611311238279901114115973762, −13.03534796539154365475424021620, −11.90762411498289592128439661582, −11.34307422856977343814786780232, −10.5074036560446536633508036020, −10.09349913203241713746409011527, −9.07294133087966638522891204386, −8.47685675145582985406623788982, −7.62808639046365003764498937302, −6.81021339243754930073965378070, −6.04628062481726093059264571635, −4.98907904356990121187273578970, −3.86800459761638059921140050225, −2.91026190170334561080178811822, −2.27776516849211280649150387433, −1.58706618814406798447616473735, 0.16907600714659358822946683144, 1.28385055233957295590402760405, 1.86317915950103074044393062687, 3.36676939295005005574244197242, 4.52116373892269180727456761768, 5.11998595100044285118305185731, 6.0940079336347070958403960137, 6.72217496258773784717049086423, 7.75839923131888254949624147022, 8.34911339253053259491927654135, 9.01558896885030647313567819811, 9.743196798631265206952471294967, 10.67886661010011822359074389459, 11.132232730629340034931731264009, 12.15549624911488896155104251542, 13.28278690791927618286470268345, 13.896884480685744056460038667100, 14.30528723093180754432583447700, 15.60675291385112359878029806611, 16.26175523754756377195922210431, 16.547541859897564436258354018925, 17.41512845305630731011218629810, 17.97148084620450695144582388720, 18.83019117749854957994385758946, 19.4874773434245016663415547709

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