↑

Properties

Label 1-1984-1984.451-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.430 - 0.902i$
Analytic cond. $9.21365$
Root an. cond. $9.21365$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.284 + 0.958i)3-s + (0.608 − 0.793i)5-s + (0.838 − 0.544i)7-s + (−0.838 − 0.544i)9-s + (0.333 − 0.942i)11-s + (−0.725 + 0.688i)13-s + (0.587 + 0.809i)15-s + (0.743 − 0.669i)17-s + (−0.0261 − 0.999i)19-s + (0.284 + 0.958i)21-s + (−0.891 + 0.453i)23-s + (−0.258 − 0.965i)25-s + (0.760 − 0.649i)27-s + (0.522 + 0.852i)29-s + (0.809 + 0.587i)33-s + ⋯
L(s)  = 1  + (−0.284 + 0.958i)3-s + (0.608 − 0.793i)5-s + (0.838 − 0.544i)7-s + (−0.838 − 0.544i)9-s + (0.333 − 0.942i)11-s + (−0.725 + 0.688i)13-s + (0.587 + 0.809i)15-s + (0.743 − 0.669i)17-s + (−0.0261 − 0.999i)19-s + (0.284 + 0.958i)21-s + (−0.891 + 0.453i)23-s + (−0.258 − 0.965i)25-s + (0.760 − 0.649i)27-s + (0.522 + 0.852i)29-s + (0.809 + 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.304230697 - 0.8225926854i\)
\(L(\frac12)\) \(\approx\) \(1.304230697 - 0.8225926854i\)
\(L(1)\) \(\approx\) \(1.108935171 - 0.09864375372i\)
\(L(1)\) \(\approx\) \(1.108935171 - 0.09864375372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.284 + 0.958i)T \)
5 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 + (0.838 - 0.544i)T \)
11 \( 1 + (0.333 - 0.942i)T \)
13 \( 1 + (-0.725 + 0.688i)T \)
17 \( 1 + (0.743 - 0.669i)T \)
19 \( 1 + (-0.0261 - 0.999i)T \)
23 \( 1 + (-0.891 + 0.453i)T \)
29 \( 1 + (0.522 + 0.852i)T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (0.777 - 0.629i)T \)
43 \( 1 + (-0.688 + 0.725i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.824 - 0.566i)T \)
59 \( 1 + (-0.878 + 0.477i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (-0.130 - 0.991i)T \)
71 \( 1 + (0.544 - 0.838i)T \)
73 \( 1 + (-0.998 + 0.0523i)T \)
79 \( 1 + (0.743 - 0.669i)T \)
83 \( 1 + (-0.878 - 0.477i)T \)
89 \( 1 + (0.891 + 0.453i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.04214989796205577138976494734, −19.139917895896647137410064807753, −18.54109125993417981817068257248, −17.94285190240397474731704886285, −17.349410845085327753066569479903, −16.89758511771430870110991641704, −15.5147856605095449825911270395, −14.67596331312570434222808282270, −14.38819332216101806933951736428, −13.559226805731724517326018148427, −12.434090012525892276459255833490, −12.20661826537018644157742896884, −11.36348288808739615305094016587, −10.29023622785850445286760554746, −9.98974203701574452423362608723, −8.647143797504377882629087599601, −7.894007752575174143564397512267, −7.32698467204453137552897225466, −6.399988362366692653590728189997, −5.73285789613913125858518294688, −5.0914612430171269967863794930, −3.846145509657264333476639181801, −2.579990195806147199818961656403, −2.07647010003877347968629859955, −1.28679200420998845221619066001, 0.557729785886491076495081760971, 1.56374008543630346310760038104, 2.74718552501859847852427242544, 3.80497880901938890923999975548, 4.63324821697885072019265665041, 5.1458469438188937212325557466, 5.88204262950926349050734600470, 6.903373625956142888518735008105, 7.95151388733844151220148857001, 8.85955053532513944553072111881, 9.31193942075102756658135918335, 10.16082777357362112059363894735, 10.86732880992556515513452225149, 11.71980934394869728428966280182, 12.166949165284746795234108984361, 13.47523306045864396362608205731, 14.117102944924812300233955407122, 14.49224438534287307193742565443, 15.66747451562769167558126323511, 16.37686478469008449405259443296, 16.79865083678211250288568500169, 17.53900381166938369596825958797, 18.03921193594350845897867337501, 19.35692658317203650873167899503, 19.98504734256332584039814725130

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