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Properties

Label 1-2001-2001.620-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.967 + 0.253i$
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  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (0.433 + 0.900i)8-s + (0.974 + 0.222i)10-s + (−0.433 + 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.974 − 0.222i)19-s + (−0.900 + 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.222 + 0.974i)25-s + (−0.974 + 0.222i)26-s + ⋯
L(s)  = 1  + (−0.781 + 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)7-s + (0.433 + 0.900i)8-s + (0.974 + 0.222i)10-s + (−0.433 + 0.900i)11-s + (0.900 + 0.433i)13-s + (−0.781 − 0.623i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.974 − 0.222i)19-s + (−0.900 + 0.433i)20-s + (−0.222 − 0.974i)22-s + (−0.222 + 0.974i)25-s + (−0.974 + 0.222i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.967 + 0.253i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (620, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.967 + 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07978862239 + 0.6201128068i\)
\(L(\frac12)\) \(\approx\) \(0.07978862239 + 0.6201128068i\)
\(L(1)\) \(\approx\) \(0.6080021059 + 0.2043928911i\)
\(L(1)\) \(\approx\) \(0.6080021059 + 0.2043928911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (-0.623 - 0.781i)T \)
7 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (-0.433 + 0.900i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.974 - 0.222i)T \)
31 \( 1 + (0.781 - 0.623i)T \)
37 \( 1 + (0.433 + 0.900i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.781 + 0.623i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.974 - 0.222i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.781 + 0.623i)T \)
79 \( 1 + (-0.433 - 0.900i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.781 - 0.623i)T \)
97 \( 1 + (0.974 + 0.222i)T \)
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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

−19.519710930653637547479749594164, −18.75755085104707257798278649730, −18.110455508362449714972581549249, −17.41913099638473485664233105909, −16.59890444820667029301409300084, −15.9925077229445937843177095886, −15.171622192365612965963536668870, −14.259032713131434913679332384925, −13.37416044080808401756008799392, −12.80018076698676435194984784513, −11.734081218099949134302124767174, −11.02725389463555629787699380039, −10.56586285620383159340387588840, −10.15242126438922527555473852971, −8.73253251584485698470698647612, −8.219537130369355693098014732209, −7.629617608839571730004335591864, −6.706089407339202071626529184140, −6.00332816882446952779682496962, −4.43302380670983100708714772803, −3.70490982866820854093170115455, −3.18681004357566190846454354814, −2.09753016921106847085823067028, −0.99357853891680351432101316182, −0.19599849721323678655881936238, 0.89849205335874190834574074904, 1.86673429567303823328826921674, 2.742482122377030894856725168584, 4.33864970840809104306394040433, 4.8316056821805446982823286899, 5.75247482616960441742817392823, 6.521142892062161061445082731234, 7.52453396965288705471607431463, 8.10969509896377507210651053545, 8.925309873663989874149910830324, 9.28820554508382275905400396144, 10.29429542443639090340841989119, 11.2911630450741818366195336922, 11.79954746593644769943298823716, 12.702855168502902948144140435511, 13.53568085621500278981434639373, 14.55264596905564444810926365027, 15.35393526281676343624906856534, 15.70646089386088320219321739170, 16.36940425630725529235380230821, 17.23428051204149657052096111510, 17.88313979576737548652844146072, 18.70103330426332331061588175229, 19.08439886600593412871440328252, 20.04930661834381817983073140644

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