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Properties

Label 1-2001-2001.122-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.990 + 0.139i$
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  + (−0.933 + 0.359i)2-s + (0.742 − 0.670i)4-s + (0.882 + 0.470i)5-s + (0.992 − 0.122i)7-s + (−0.452 + 0.891i)8-s + (−0.992 − 0.122i)10-s + (−0.685 − 0.728i)11-s + (0.947 − 0.320i)13-s + (−0.882 + 0.470i)14-s + (0.101 − 0.994i)16-s + (0.959 + 0.281i)17-s + (0.742 − 0.670i)19-s + (0.970 − 0.242i)20-s + (0.900 + 0.433i)22-s + (0.557 + 0.830i)25-s + (−0.768 + 0.639i)26-s + ⋯
L(s)  = 1  + (−0.933 + 0.359i)2-s + (0.742 − 0.670i)4-s + (0.882 + 0.470i)5-s + (0.992 − 0.122i)7-s + (−0.452 + 0.891i)8-s + (−0.992 − 0.122i)10-s + (−0.685 − 0.728i)11-s + (0.947 − 0.320i)13-s + (−0.882 + 0.470i)14-s + (0.101 − 0.994i)16-s + (0.959 + 0.281i)17-s + (0.742 − 0.670i)19-s + (0.970 − 0.242i)20-s + (0.900 + 0.433i)22-s + (0.557 + 0.830i)25-s + (−0.768 + 0.639i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.990 + 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.591340007 + 0.1116341693i\)
\(L(\frac12)\) \(\approx\) \(1.591340007 + 0.1116341693i\)
\(L(1)\) \(\approx\) \(1.013656235 + 0.1106453721i\)
\(L(1)\) \(\approx\) \(1.013656235 + 0.1106453721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.933 + 0.359i)T \)
5 \( 1 + (0.882 + 0.470i)T \)
7 \( 1 + (0.992 - 0.122i)T \)
11 \( 1 + (-0.685 - 0.728i)T \)
13 \( 1 + (0.947 - 0.320i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.742 - 0.670i)T \)
31 \( 1 + (-0.794 + 0.607i)T \)
37 \( 1 + (0.182 - 0.983i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.794 + 0.607i)T \)
47 \( 1 + (0.623 - 0.781i)T \)
53 \( 1 + (-0.714 - 0.699i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (0.0203 - 0.999i)T \)
67 \( 1 + (-0.685 + 0.728i)T \)
71 \( 1 + (0.862 + 0.505i)T \)
73 \( 1 + (0.0611 + 0.998i)T \)
79 \( 1 + (0.101 + 0.994i)T \)
83 \( 1 + (0.917 + 0.396i)T \)
89 \( 1 + (0.979 + 0.202i)T \)
97 \( 1 + (0.262 - 0.965i)T \)
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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

−20.3102020439504390267171236469, −18.85657711463167093066965942296, −18.43552491505979679077589473493, −17.87673438453065374050787612240, −17.16103650086178226126876074811, −16.52635870725758264032212988564, −15.781839353468773284833606657, −14.912544431329112985402923215716, −13.96835128921808778474975625889, −13.29119221287844585622204732062, −12.26539119751855645789957404384, −11.882372086919824601134783433965, −10.74384964609284307565831047079, −10.34526252691199515288210884657, −9.36818623822859949980187911942, −8.92505420000851310844322139345, −7.847284522925908113335593881134, −7.5591909715206377006938720359, −6.26072682171261415772802137419, −5.5357909279806932013094732003, −4.64655002497322263760831902351, −3.52466548347277752728404526432, −2.43788508861250952181493948722, −1.668915394165251249665531893950, −1.06585267758514967927842966040, 0.9161734497591350565692989628, 1.64926031708462872602232516967, 2.63474288392924204151037062655, 3.49031201035168422985423772361, 5.15334724765840273132560466019, 5.54753986937240795095372032027, 6.35190956253914437526258572336, 7.30622304290861871536474885585, 7.98830612333468273144099959214, 8.670698098236972146636841467624, 9.50122212276802180469293529477, 10.31313128968866814483394456828, 11.0036497731879525435573420780, 11.30765532762019521526509315895, 12.613334562461068303563411442736, 13.66192536356474686185680982340, 14.18978805789638262977894744740, 14.8775276445928419466657888050, 15.74555413482494240720267090004, 16.4126214040425362978256206897, 17.23651290198524581889056518286, 17.89293060773598922257441551164, 18.36321039023780139311459583778, 18.87225366647307094445182289576, 19.959507293127478020244019182422

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