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Properties

Label 1-2001-2001.1655-r1-0-0
Degree $1$
Conductor $2001$
Sign $-0.991 - 0.129i$
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  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (−0.781 − 0.623i)8-s + (−0.433 + 0.900i)10-s + (0.781 − 0.623i)11-s + (−0.623 − 0.781i)13-s + (−0.974 + 0.222i)14-s + (0.623 + 0.781i)16-s − i·17-s + (0.433 − 0.900i)19-s + (0.623 − 0.781i)20-s + (−0.900 + 0.433i)22-s + (−0.900 − 0.433i)25-s + (0.433 + 0.900i)26-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (0.222 − 0.974i)5-s + (0.900 − 0.433i)7-s + (−0.781 − 0.623i)8-s + (−0.433 + 0.900i)10-s + (0.781 − 0.623i)11-s + (−0.623 − 0.781i)13-s + (−0.974 + 0.222i)14-s + (0.623 + 0.781i)16-s − i·17-s + (0.433 − 0.900i)19-s + (0.623 − 0.781i)20-s + (−0.900 + 0.433i)22-s + (−0.900 − 0.433i)25-s + (0.433 + 0.900i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.991 - 0.129i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ -0.991 - 0.129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1149421201 - 1.764600097i\)
\(L(\frac12)\) \(\approx\) \(0.1149421201 - 1.764600097i\)
\(L(1)\) \(\approx\) \(0.7332857230 - 0.5023277141i\)
\(L(1)\) \(\approx\) \(0.7332857230 - 0.5023277141i\)

Euler product

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

−19.86276185061584939340926847052, −19.175280733252192736402683473432, −18.6643323556734232898212973458, −17.88539355473373470074831130367, −17.31833644561661317180214119643, −16.80673113436415593855268217505, −15.660555948051993821540716463764, −14.99094298853253917208971126579, −14.484952304489435420383433975701, −13.95710877300246456341335701808, −12.36898558840366595752034776505, −11.80781006223654824968094743529, −11.15544913181631300729341256010, −10.3161132367440648947981706466, −9.71034248078309210309515742857, −8.95680555167265617037613104878, −8.00731363580408164051148497446, −7.45505605688512970878450679862, −6.52990650122778841788910619883, −6.047607504274611645357392878457, −4.94562660102988935317057049133, −3.872034819128685119079669717230, −2.67570620871750734990766198396, −1.902631996311522109178451186265, −1.29152226348005114489818822400, 0.53530884704622940492530294506, 0.84266761558415940109950913820, 1.89901038397528874010135494344, 2.83010687210380820821477015915, 3.926110712377221910521342132621, 4.9227887240648951298834946807, 5.636924957054518875766097870532, 6.8009250191377945535054401197, 7.55993227765128278410346129725, 8.23749415857602103068283253114, 9.03898722298668786571517211674, 9.48559310988025453432706302704, 10.51462936079969867017188532987, 11.15537731261992659851363193434, 12.00652754071589969195532289556, 12.43032835662491150403786240240, 13.667037127900626281251794105918, 14.11383173667903079328600512129, 15.397978113668121835664685770275, 15.85904026667419553208643978127, 16.88013615140562352796740460295, 17.253862173171097795055097544182, 17.76809760446450484588907764644, 18.645050801051801744054279362101, 19.59624381432916398191135487035

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