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Properties

Label 1-2001-2001.1586-r0-0-0
Degree $1$
Conductor $2001$
Sign $0.546 - 0.837i$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)5-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + 17-s + (0.222 + 0.974i)19-s + (−0.900 − 0.433i)20-s + (0.222 − 0.974i)22-s + (−0.222 − 0.974i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.623 − 0.781i)5-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (0.222 + 0.974i)10-s + (−0.900 + 0.433i)11-s + (−0.900 + 0.433i)13-s + (0.623 + 0.781i)14-s + (−0.900 + 0.433i)16-s + 17-s + (0.222 + 0.974i)19-s + (−0.900 − 0.433i)20-s + (0.222 − 0.974i)22-s + (−0.222 − 0.974i)25-s + (0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (1586, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ 0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9225666130 - 0.4999014903i\)
\(L(\frac12)\) \(\approx\) \(0.9225666130 - 0.4999014903i\)
\(L(1)\) \(\approx\) \(0.8223096071 + 0.002819163841i\)
\(L(1)\) \(\approx\) \(0.8223096071 + 0.002819163841i\)

Euler product

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

−19.92905152851872403338573911608, −19.23158272259005380848660479884, −18.48505777451296936712172889873, −18.15617023575665432531267569682, −17.423890590019521565996665373875, −16.67109377335546053801482202843, −15.670459683359692273366793066492, −15.013134515771756683524834855625, −14.07564545877920147122696226146, −13.35093718768368485073922426637, −12.527701503399407624611924587610, −11.88019764717282297586252624065, −11.03958377228472475238261937732, −10.41174383200304827890011167643, −9.71352523879106547327967809832, −9.067539531673908868396650151722, −8.08095036087007695883192097014, −7.5225553072556625110372892691, −6.51453322569106239743290781831, −5.458170225907264703772878533324, −4.86489263783951866514117486087, −3.37224631286275373848707580410, −2.706607465382661029122061567376, −2.30436950636196371143365415582, −1.05447220102206796860860253286, 0.50471299985450079125601906844, 1.49312826916165330901566680029, 2.30379023201325341370158793278, 3.87443371241553268383025729706, 4.86982001820324998265818327159, 5.26965592719827519362494117000, 6.2336096459264143955758537705, 7.15091591480970426847924398818, 7.863228084299530847334409604111, 8.35149231150261940085986198847, 9.5925748080323852126322110919, 9.93522175025766104985837641911, 10.48834380607483058848281865462, 11.69296621316690508029412423080, 12.5906801250162332886582429661, 13.4793747374848590149395506578, 14.02381223791036037656908291910, 14.74192064144332174339948713044, 15.58341548006054655221315830841, 16.55463045915684554435802648514, 16.87842255981520188946886870557, 17.37984926859603923716706603749, 18.34937103194665600678086773286, 18.83579447906352427266152517910, 19.94052347104994909632697969636

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