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Properties

Label 1-2001-2001.1019-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.865 + 0.501i$
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  + (−0.301 + 0.953i)2-s + (−0.818 − 0.574i)4-s + (0.917 − 0.396i)5-s + (−0.101 + 0.994i)7-s + (0.794 − 0.607i)8-s + (0.101 + 0.994i)10-s + (0.933 + 0.359i)11-s + (−0.714 + 0.699i)13-s + (−0.917 − 0.396i)14-s + (0.339 + 0.940i)16-s + (0.959 + 0.281i)17-s + (−0.818 − 0.574i)19-s + (−0.979 − 0.202i)20-s + (−0.623 + 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.452 − 0.891i)26-s + ⋯
L(s)  = 1  + (−0.301 + 0.953i)2-s + (−0.818 − 0.574i)4-s + (0.917 − 0.396i)5-s + (−0.101 + 0.994i)7-s + (0.794 − 0.607i)8-s + (0.101 + 0.994i)10-s + (0.933 + 0.359i)11-s + (−0.714 + 0.699i)13-s + (−0.917 − 0.396i)14-s + (0.339 + 0.940i)16-s + (0.959 + 0.281i)17-s + (−0.818 − 0.574i)19-s + (−0.979 − 0.202i)20-s + (−0.623 + 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.452 − 0.891i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.865 + 0.501i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3342943343 + 1.244281928i\)
\(L(\frac12)\) \(\approx\) \(0.3342943343 + 1.244281928i\)
\(L(1)\) \(\approx\) \(0.7963651599 + 0.5659959321i\)
\(L(1)\) \(\approx\) \(0.7963651599 + 0.5659959321i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.301 + 0.953i)T \)
5 \( 1 + (0.917 - 0.396i)T \)
7 \( 1 + (-0.101 + 0.994i)T \)
11 \( 1 + (0.933 + 0.359i)T \)
13 \( 1 + (-0.714 + 0.699i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (-0.818 - 0.574i)T \)
31 \( 1 + (-0.0203 + 0.999i)T \)
37 \( 1 + (-0.591 + 0.806i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.0203 + 0.999i)T \)
47 \( 1 + (-0.222 + 0.974i)T \)
53 \( 1 + (-0.992 + 0.122i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.970 + 0.242i)T \)
67 \( 1 + (0.933 - 0.359i)T \)
71 \( 1 + (-0.996 - 0.0815i)T \)
73 \( 1 + (-0.742 + 0.670i)T \)
79 \( 1 + (0.339 - 0.940i)T \)
83 \( 1 + (0.182 - 0.983i)T \)
89 \( 1 + (0.768 - 0.639i)T \)
97 \( 1 + (-0.999 - 0.0407i)T \)
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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

−19.55628592024319128239750645266, −19.06219956659066415030684798069, −18.29962161148833603861194618291, −17.322199762075951074881650197086, −17.10583235997813253749080911316, −16.45802138085809754055259881166, −14.959978320825699134457479217651, −14.21647829924160334912169792795, −13.80441479412837622273297959914, −12.936467441730266300183676478507, −12.28435493718291385464716109498, −11.38356732356040639314948563892, −10.53072474718456411297975840754, −10.11380362797601912897658022177, −9.48317228244757238360130767748, −8.58964496180341965907300158228, −7.63785469595268114930166719472, −6.91839738411359754742198149072, −5.855430081602699294587843285813, −5.02020455700911329944702594568, −3.88944646867442322057244830694, −3.3755631757257569922444269702, −2.32246187349099566575167578204, −1.517575419042263527096082119687, −0.50645091243953425929147375927, 1.306327907185035088807898498632, 1.969246470838385374269300287597, 3.227078543366211319002686270538, 4.67661060426708462099634978433, 4.93502022482464839875045586885, 6.15124074601551887384878825666, 6.36773612852320461444301509874, 7.35895810918272558860744613820, 8.473879791487254762555333516031, 8.951124536162853580728626821274, 9.667276581838343899256152672771, 10.1354941511348124703110830103, 11.4300778056926197774579704923, 12.4159160734198809967606626373, 12.8773310446647543400415088958, 13.96509757031650250560071566258, 14.517177872997850278403193805484, 15.035673551674547513545406671860, 16.03597508153702271943849526905, 16.67350703292237940541465367956, 17.36950243399516283713927463705, 17.75088058362246944661858828804, 18.8290686897313479134073297979, 19.191224759941330216910865500034, 20.09693512870762473711538549446

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