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Properties

Label 1-2001-2001.1211-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.884 - 0.467i$
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  + (0.986 − 0.162i)2-s + (0.947 − 0.320i)4-s + (−0.301 − 0.953i)5-s + (0.452 − 0.891i)7-s + (0.882 − 0.470i)8-s + (−0.452 − 0.891i)10-s + (−0.488 − 0.872i)11-s + (−0.979 − 0.202i)13-s + (0.301 − 0.953i)14-s + (0.794 − 0.607i)16-s + (−0.841 + 0.540i)17-s + (0.947 − 0.320i)19-s + (−0.591 − 0.806i)20-s + (−0.623 − 0.781i)22-s + (−0.818 + 0.574i)25-s + (−0.999 − 0.0407i)26-s + ⋯
L(s)  = 1  + (0.986 − 0.162i)2-s + (0.947 − 0.320i)4-s + (−0.301 − 0.953i)5-s + (0.452 − 0.891i)7-s + (0.882 − 0.470i)8-s + (−0.452 − 0.891i)10-s + (−0.488 − 0.872i)11-s + (−0.979 − 0.202i)13-s + (0.301 − 0.953i)14-s + (0.794 − 0.607i)16-s + (−0.841 + 0.540i)17-s + (0.947 − 0.320i)19-s + (−0.591 − 0.806i)20-s + (−0.623 − 0.781i)22-s + (−0.818 + 0.574i)25-s + (−0.999 − 0.0407i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.884 - 0.467i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.884 - 0.467i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5938050784 - 2.394186499i\)
\(L(\frac12)\) \(\approx\) \(0.5938050784 - 2.394186499i\)
\(L(1)\) \(\approx\) \(1.452817894 - 0.9331986775i\)
\(L(1)\) \(\approx\) \(1.452817894 - 0.9331986775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.986 - 0.162i)T \)
5 \( 1 + (-0.301 - 0.953i)T \)
7 \( 1 + (0.452 - 0.891i)T \)
11 \( 1 + (-0.488 - 0.872i)T \)
13 \( 1 + (-0.979 - 0.202i)T \)
17 \( 1 + (-0.841 + 0.540i)T \)
19 \( 1 + (0.947 - 0.320i)T \)
31 \( 1 + (-0.917 - 0.396i)T \)
37 \( 1 + (0.996 + 0.0815i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.917 - 0.396i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.768 - 0.639i)T \)
59 \( 1 + (-0.415 + 0.909i)T \)
61 \( 1 + (0.182 - 0.983i)T \)
67 \( 1 + (-0.488 + 0.872i)T \)
71 \( 1 + (0.0611 - 0.998i)T \)
73 \( 1 + (0.523 + 0.852i)T \)
79 \( 1 + (0.794 + 0.607i)T \)
83 \( 1 + (-0.862 - 0.505i)T \)
89 \( 1 + (-0.262 + 0.965i)T \)
97 \( 1 + (0.685 + 0.728i)T \)
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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

−20.34274248067198755472626952893, −19.70406714104271566911457650914, −18.803240012612615196150537159657, −18.014042199451595024859068924038, −17.48824351688363177375527485855, −16.276980177229331045663330678061, −15.6005983424268335999663846608, −15.108186389286109643429564991275, −14.41327311483978139216124444474, −13.93810595733856631273046840031, −12.74671473279944063138507239201, −12.257972032335905176229477202588, −11.476105296932828527493339782, −10.94900267019810041236150494528, −9.96784982607164430723045434053, −9.10023513851329473611853662083, −7.70237629816109436608113048572, −7.501962930325989857054455178314, −6.59576909581020506618579856679, −5.71708384612325795254328988586, −4.9264697558295199235143516803, −4.284786691383163200963766962126, −3.10451108068407257756603147129, −2.49948578307351015036814143176, −1.83431193355119477536406027671, 0.54191927572324015162282167363, 1.51626030904361290470906098971, 2.562874416167932995327994416195, 3.58480994726552866581685114022, 4.31301773482910408960692446295, 5.02147333038437404830551399841, 5.61060341017918390280313846105, 6.699915471952586875471390122161, 7.61977279806248347370005680282, 8.0765994885000149477423169592, 9.25800888437363428536296218917, 10.14187807870373542833715286142, 11.0634716220231046892907333342, 11.48046052977831316967989008784, 12.42061622198387393109060199679, 13.15835557673153287014865725501, 13.55751556844054871044654537430, 14.44095956317877020964034572796, 15.177361285758325303906822470223, 15.98307788608438994928122897703, 16.60261990803125791587470824230, 17.185278180859255487715421323564, 18.16921809049058250617220817761, 19.33149146764272387329084187113, 19.94195968794245022548800831277

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