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Properties

Label 1-2001-2001.68-r1-0-0
Degree $1$
Conductor $2001$
Sign $0.853 - 0.521i$
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  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (−0.781 + 0.623i)10-s + (−0.974 − 0.222i)11-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.781 − 0.623i)19-s + (−0.222 − 0.974i)20-s + (0.623 − 0.781i)22-s + (0.623 + 0.781i)25-s + (0.781 + 0.623i)26-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (0.900 + 0.433i)5-s + (−0.623 + 0.781i)7-s + (0.974 − 0.222i)8-s + (−0.781 + 0.623i)10-s + (−0.974 − 0.222i)11-s + (0.222 − 0.974i)13-s + (−0.433 − 0.900i)14-s + (−0.222 + 0.974i)16-s + i·17-s + (0.781 − 0.623i)19-s + (−0.222 − 0.974i)20-s + (0.623 − 0.781i)22-s + (0.623 + 0.781i)25-s + (0.781 + 0.623i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $0.853 - 0.521i$
Analytic conductor: \(215.037\)
Root analytic conductor: \(215.037\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (1:\ ),\ 0.853 - 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8278244453 - 0.2329811686i\)
\(L(\frac12)\) \(\approx\) \(0.8278244453 - 0.2329811686i\)
\(L(1)\) \(\approx\) \(0.7309311367 + 0.3342661860i\)
\(L(1)\) \(\approx\) \(0.7309311367 + 0.3342661860i\)

Euler product

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

−19.97199304366897271540890025972, −19.08554499877302927809822852558, −18.37817011957354043115612231004, −17.836060292377201418440077151772, −16.99730953653197922545898927634, −16.36387058666285859626533148138, −15.86746657029423146306628930463, −14.21294452723009787488204657119, −13.7835877956948202175422056463, −13.21012886744614389267146782792, −12.44058310063495745523709417335, −11.73633930499264127392447858694, −10.6780667956731405455589502520, −10.19493712482224084162799570068, −9.44791070058161977386724465650, −8.99060641906141606062660831210, −7.8484755814998997238187942224, −7.18391954743028538026836657694, −6.167565929754935414189303914003, −5.05787053848688919147757385979, −4.44747925155352718869148858120, −3.354468395543457390760256771575, −2.60802701145251991844583849937, −1.64611774061397321250668971372, −0.84175555981594841267886593079, 0.21030237316840513912497819485, 1.39160480332340437746029015541, 2.54890432852324000897086093061, 3.24082124013047695311213446822, 4.70701637711925740400370313348, 5.57604325157391371935795648129, 6.002696404766805549656915433308, 6.6604565357530237732868374981, 7.80021451113924698911776083901, 8.27301803126336924329895040974, 9.39013836932685550949218574200, 9.756402293764554062211084501797, 10.57486311936257639224077251050, 11.27642033524166328792297865991, 12.907094943096793067585785107432, 13.00688474816278119774141845600, 13.91010911465612523718371446735, 14.85546259206890996023462332785, 15.34921333068147011209256849722, 15.99958560869943448081290098405, 16.80768474405598962576964672282, 17.651564664128451226047826580256, 18.17730846392703623317001221517, 18.65531101638524742876257942149, 19.47786753853284487888452728675

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