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Properties

Label 1-2001-2001.1295-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.989 - 0.147i$
Analytic cond. $215.037$
Root an. cond. $215.037$
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  + (−0.953 − 0.301i)2-s + (0.818 + 0.574i)4-s + (−0.917 + 0.396i)5-s + (−0.101 + 0.994i)7-s + (−0.607 − 0.794i)8-s + (0.994 − 0.101i)10-s + (−0.359 + 0.933i)11-s + (0.714 − 0.699i)13-s + (0.396 − 0.917i)14-s + (0.339 + 0.940i)16-s + (−0.281 + 0.959i)17-s + (0.574 − 0.818i)19-s + (−0.979 − 0.202i)20-s + (0.623 − 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.891 + 0.452i)26-s + ⋯
L(s)  = 1  + (−0.953 − 0.301i)2-s + (0.818 + 0.574i)4-s + (−0.917 + 0.396i)5-s + (−0.101 + 0.994i)7-s + (−0.607 − 0.794i)8-s + (0.994 − 0.101i)10-s + (−0.359 + 0.933i)11-s + (0.714 − 0.699i)13-s + (0.396 − 0.917i)14-s + (0.339 + 0.940i)16-s + (−0.281 + 0.959i)17-s + (0.574 − 0.818i)19-s + (−0.979 − 0.202i)20-s + (0.623 − 0.781i)22-s + (0.685 − 0.728i)25-s + (−0.891 + 0.452i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.989 - 0.147i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03638142063 + 0.4897748924i\)
\(L(\frac12)\) \(\approx\) \(-0.03638142063 + 0.4897748924i\)
\(L(1)\) \(\approx\) \(0.5488053109 + 0.1391006662i\)
\(L(1)\) \(\approx\) \(0.5488053109 + 0.1391006662i\)

Euler product

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

−19.375828695173126154173273059553, −18.5465868374660447788767875447, −18.175044403350333827646381218147, −16.92924358388645919955938480936, −16.45422448440642497765359966507, −16.074219210122121905872122311237, −15.33558165249932340038587819966, −14.2113381049852134065514040524, −13.75765409048520074900330705241, −12.64629230346553177852689780590, −11.67496416377852936763292428541, −11.09407818018935940948549438754, −10.59052093983754737192978161225, −9.44285194017965388239697147848, −8.93039765984743888660593537660, −7.9515343799665759586711458931, −7.56181035743963548820524796309, −6.730919291339356597996907645274, −5.84267456661963973142566141177, −4.86929575286609070651843793289, −3.821532710509210345602120961600, −3.11904398794781010054445674555, −1.72405427890110239243061569160, −0.773020625669364076259890790073, −0.178550158374330549181582982238, 1.03942595631492445549894512943, 2.16160309282590754070837858509, 2.92190234389204606063703796513, 3.671666121489379277249264617695, 4.76595376685093738473700929195, 5.947165559101870989550007976061, 6.69713043187238612860645119796, 7.632558218996039843764835716914, 8.11808141631735545977951286503, 8.93677493511299549830079204409, 9.64955005805936420725245560542, 10.623086080566031100949248091402, 11.12586467527581216656960622877, 11.908586804749845539981716806068, 12.59481205337467479599029507914, 13.172855100213033239895377141320, 14.7494998494177712858413031997, 15.29977845938811374586038548478, 15.69141522904576086621713880405, 16.46895771400111590514586394812, 17.529731011925425742975595323912, 18.155367453212733952427687336793, 18.56685136018093695960179737464, 19.3979162024116532692545312342, 20.06418919923639325791972112639

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