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Properties

Label 1-2001-2001.32-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.995 - 0.0970i$
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  + (−0.852 − 0.523i)2-s + (0.452 + 0.891i)4-s + (0.742 − 0.670i)5-s + (0.182 − 0.983i)7-s + (0.0815 − 0.996i)8-s + (−0.983 + 0.182i)10-s + (0.940 + 0.339i)11-s + (−0.882 − 0.470i)13-s + (−0.670 + 0.742i)14-s + (−0.591 + 0.806i)16-s + (−0.909 + 0.415i)17-s + (−0.891 + 0.452i)19-s + (0.933 + 0.359i)20-s + (−0.623 − 0.781i)22-s + (0.101 − 0.994i)25-s + (0.505 + 0.862i)26-s + ⋯
L(s)  = 1  + (−0.852 − 0.523i)2-s + (0.452 + 0.891i)4-s + (0.742 − 0.670i)5-s + (0.182 − 0.983i)7-s + (0.0815 − 0.996i)8-s + (−0.983 + 0.182i)10-s + (0.940 + 0.339i)11-s + (−0.882 − 0.470i)13-s + (−0.670 + 0.742i)14-s + (−0.591 + 0.806i)16-s + (−0.909 + 0.415i)17-s + (−0.891 + 0.452i)19-s + (0.933 + 0.359i)20-s + (−0.623 − 0.781i)22-s + (0.101 − 0.994i)25-s + (0.505 + 0.862i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.995 - 0.0970i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.995 - 0.0970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03584744404 - 0.7369193794i\)
\(L(\frac12)\) \(\approx\) \(0.03584744404 - 0.7369193794i\)
\(L(1)\) \(\approx\) \(0.6401602041 - 0.3677465656i\)
\(L(1)\) \(\approx\) \(0.6401602041 - 0.3677465656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.852 - 0.523i)T \)
5 \( 1 + (0.742 - 0.670i)T \)
7 \( 1 + (0.182 - 0.983i)T \)
11 \( 1 + (0.940 + 0.339i)T \)
13 \( 1 + (-0.882 - 0.470i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (-0.891 + 0.452i)T \)
31 \( 1 + (0.830 - 0.557i)T \)
37 \( 1 + (0.872 + 0.488i)T \)
41 \( 1 + (0.281 + 0.959i)T \)
43 \( 1 + (-0.830 - 0.557i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.917 - 0.396i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.728 - 0.685i)T \)
67 \( 1 + (-0.339 - 0.940i)T \)
71 \( 1 + (-0.714 - 0.699i)T \)
73 \( 1 + (-0.639 - 0.768i)T \)
79 \( 1 + (-0.806 + 0.591i)T \)
83 \( 1 + (0.818 - 0.574i)T \)
89 \( 1 + (0.953 - 0.301i)T \)
97 \( 1 + (-0.925 - 0.377i)T \)
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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

−20.01472360280735976223590200771, −19.23530339962094789749806822014, −18.92156285844799059093696584845, −17.82757488405678974089211314691, −17.6602845651576011550330296738, −16.80239508149373810510079652248, −16.013290471053779421972109362096, −15.09557046718900638986338316186, −14.66624812502130776028238652139, −14.03469238228350274168287294292, −13.07252922655205068696242872712, −11.849641425718244269961452429875, −11.37318771025529687660254439480, −10.51652601974344552479040225059, −9.67021248236562586497746215651, −9.06531252527817452977061063518, −8.56715815310529741218519692935, −7.397331533096423616392170739524, −6.60792609853519828699872908285, −6.217243639499382953689812169888, −5.28124109219526417253247235178, −4.41296992554251782737574183866, −2.78363080643561092674354968581, −2.27803444415519091778321275286, −1.38019463476697005326661922031, 0.32099508157382839069292525623, 1.46222281284274563172126394172, 1.99207759951685845115333117171, 3.12742513113386753437019541853, 4.28488676968167456461831681121, 4.6868817793326922443263897469, 6.254140438506657903569110914242, 6.70614852808190394561474204042, 7.83090989203702814764083519561, 8.35600815577902980724601633845, 9.35189086141951169130004603134, 9.850704353388183014213733032847, 10.50176030817010660997592721285, 11.34454402435287758071947678080, 12.17572140900041337011017613471, 12.934370269723450943399363845759, 13.43225689212770204957893220448, 14.48901722447448696552656223313, 15.237081974764917619881619087360, 16.48493994381076259975548613914, 16.840631327545148087518751948180, 17.5374030800573264995621429665, 17.79219631744637298411495727110, 19.041092448191086688853788640621, 19.73247959095449163323669224930

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