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Properties

Label 1-2001-2001.101-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.884 + 0.465i$
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  + (0.699 − 0.714i)2-s + (−0.0203 − 0.999i)4-s + (0.488 + 0.872i)5-s + (0.262 + 0.965i)7-s + (−0.728 − 0.685i)8-s + (0.965 + 0.262i)10-s + (0.320 + 0.947i)11-s + (−0.182 − 0.983i)13-s + (0.872 + 0.488i)14-s + (−0.999 + 0.0407i)16-s + (−0.909 + 0.415i)17-s + (0.999 − 0.0203i)19-s + (0.862 − 0.505i)20-s + (0.900 + 0.433i)22-s + (−0.523 + 0.852i)25-s + (−0.830 − 0.557i)26-s + ⋯
L(s)  = 1  + (0.699 − 0.714i)2-s + (−0.0203 − 0.999i)4-s + (0.488 + 0.872i)5-s + (0.262 + 0.965i)7-s + (−0.728 − 0.685i)8-s + (0.965 + 0.262i)10-s + (0.320 + 0.947i)11-s + (−0.182 − 0.983i)13-s + (0.872 + 0.488i)14-s + (−0.999 + 0.0407i)16-s + (−0.909 + 0.415i)17-s + (0.999 − 0.0203i)19-s + (0.862 − 0.505i)20-s + (0.900 + 0.433i)22-s + (−0.523 + 0.852i)25-s + (−0.830 − 0.557i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.884 + 0.465i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.884 + 0.465i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.363071988 + 0.5839900798i\)
\(L(\frac12)\) \(\approx\) \(2.363071988 + 0.5839900798i\)
\(L(1)\) \(\approx\) \(1.605093948 - 0.1444473182i\)
\(L(1)\) \(\approx\) \(1.605093948 - 0.1444473182i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.699 - 0.714i)T \)
5 \( 1 + (0.488 + 0.872i)T \)
7 \( 1 + (0.262 + 0.965i)T \)
11 \( 1 + (0.320 + 0.947i)T \)
13 \( 1 + (-0.182 - 0.983i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.999 - 0.0203i)T \)
31 \( 1 + (0.359 + 0.933i)T \)
37 \( 1 + (0.925 - 0.377i)T \)
41 \( 1 + (0.281 + 0.959i)T \)
43 \( 1 + (-0.359 + 0.933i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.591 - 0.806i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.953 + 0.301i)T \)
67 \( 1 + (-0.947 - 0.320i)T \)
71 \( 1 + (0.101 - 0.994i)T \)
73 \( 1 + (-0.607 + 0.794i)T \)
79 \( 1 + (-0.0407 + 0.999i)T \)
83 \( 1 + (-0.986 + 0.162i)T \)
89 \( 1 + (0.0815 + 0.996i)T \)
97 \( 1 + (0.670 + 0.742i)T \)
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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

−20.2554333493370620752886874980, −19.24676877352435076737852933422, −18.15311201272994371911958008727, −17.426988638081162503941961198475, −16.7923938946529033944085717750, −16.35126512074050463334985248627, −15.68658936825396323438945642428, −14.55586511678667914437030676898, −13.85912338818390438811708215505, −13.55721697974006043076036711937, −12.85545825289446949433567871041, −11.6293443080780442107182280182, −11.49174546215917662097426931468, −10.12794649445834093927566249711, −9.16024353543890794928051833707, −8.65317689057549058092887961333, −7.69616837321022605100096255335, −6.98280874970060973880432829558, −6.14912483968675627041251870605, −5.414909579218167528974545479, −4.46278162810980916573685253144, −4.10952091472730950209948634733, −2.95902386594251451315470742745, −1.8345056851913179454000602976, −0.64903250288208526971571230034, 1.31498117843330686103611005634, 2.21823430649295902241216995438, 2.77974212151158371910402817175, 3.624268107494859378951977758, 4.77260170143018314073424623916, 5.37570508056825474670952376926, 6.234361377930696650118737767648, 6.86604196160210348162613656102, 7.9699536249610607003335063631, 9.0986109745254683565641947407, 9.81177862733456670204873400122, 10.3577885541681804434860250974, 11.33901983621530385350961314073, 11.75512261109416333169874740148, 12.78421235516068093123238225243, 13.17813874435971494303237037989, 14.27630307668802251879926445333, 14.79432505910502515560929674470, 15.27156258872464840569244039208, 16.04087334822032069034778898977, 17.553750422726724865657493884775, 18.017518070902982926661616955268, 18.41432437400328456599888876769, 19.62827983014842119890782327374, 19.848534244184822413652823612148

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