↑

Properties

Label 1-1021-1021.130-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.900 - 0.434i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (−0.378 + 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (−0.0799 + 0.996i)5-s + (−0.903 + 0.429i)6-s + (−0.786 − 0.617i)7-s + (0.918 − 0.395i)8-s + (0.0922 + 0.995i)9-s + (−0.892 − 0.451i)10-s + (0.355 + 0.934i)11-s + (−0.0554 − 0.998i)12-s + (0.850 + 0.526i)13-s + (0.869 − 0.494i)14-s + (−0.730 + 0.682i)15-s + (0.0184 + 0.999i)16-s + (0.999 − 0.0246i)17-s + ⋯
L(s)  = 1  + (−0.378 + 0.925i)2-s + (0.739 + 0.673i)3-s + (−0.713 − 0.700i)4-s + (−0.0799 + 0.996i)5-s + (−0.903 + 0.429i)6-s + (−0.786 − 0.617i)7-s + (0.918 − 0.395i)8-s + (0.0922 + 0.995i)9-s + (−0.892 − 0.451i)10-s + (0.355 + 0.934i)11-s + (−0.0554 − 0.998i)12-s + (0.850 + 0.526i)13-s + (0.869 − 0.494i)14-s + (−0.730 + 0.682i)15-s + (0.0184 + 0.999i)16-s + (0.999 − 0.0246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2508281463 + 1.098152444i\)
\(L(\frac12)\) \(\approx\) \(-0.2508281463 + 1.098152444i\)
\(L(1)\) \(\approx\) \(0.5676704380 + 0.7620122622i\)
\(L(1)\) \(\approx\) \(0.5676704380 + 0.7620122622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.378 + 0.925i)T \)
3 \( 1 + (0.739 + 0.673i)T \)
5 \( 1 + (-0.0799 + 0.996i)T \)
7 \( 1 + (-0.786 - 0.617i)T \)
11 \( 1 + (0.355 + 0.934i)T \)
13 \( 1 + (0.850 + 0.526i)T \)
17 \( 1 + (0.999 - 0.0246i)T \)
19 \( 1 + (-0.816 - 0.577i)T \)
23 \( 1 + (-0.875 + 0.483i)T \)
29 \( 1 + (-0.927 - 0.372i)T \)
31 \( 1 + (0.999 + 0.0123i)T \)
37 \( 1 + (0.582 + 0.812i)T \)
41 \( 1 + (0.816 + 0.577i)T \)
43 \( 1 + (0.641 + 0.767i)T \)
47 \( 1 + (-0.862 - 0.505i)T \)
53 \( 1 + (-0.423 + 0.905i)T \)
59 \( 1 + (-0.949 - 0.314i)T \)
61 \( 1 + (-0.00615 + 0.999i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.918 + 0.395i)T \)
73 \( 1 + (0.869 + 0.494i)T \)
79 \( 1 + (-0.542 + 0.840i)T \)
83 \( 1 + (0.213 - 0.976i)T \)
89 \( 1 + (0.552 - 0.833i)T \)
97 \( 1 + (-0.969 - 0.243i)T \)
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

−20.92355707073259775377077749756, −20.36786137855557387362752802819, −19.44204444644288290231235116702, −19.053364062936360936125252878188, −18.38369473219826196711714777516, −17.446866313870395641882765174726, −16.48654663260397248607731816900, −15.90773851498081336895879286581, −14.57327330277804079536825865807, −13.722370100671606074267672411315, −12.97763408363806588540334651741, −12.46140504715678199828239089165, −11.877435637276692199071627623956, −10.72295642436270370673339622795, −9.61572965519658347704230650066, −9.05631349632059776774632608851, −8.27976600287473151687488613018, −7.88127698644196363015921321170, −6.30344924655648261085909406993, −5.56919288032794993216932419736, −3.972997000331509646456335527, −3.482471390188620522715973841698, −2.44817474148942883753882070480, −1.44534641320233656111677531577, −0.51876981521206488061452142743, 1.57123553935034098244014474440, 2.89210077247670606131398915654, 3.953151037773646228175977200568, 4.42141905960717691517111666055, 5.92994973865764507367921263827, 6.61929804677593399030658887464, 7.48793596103260057228522428703, 8.10838913657832655925763205403, 9.34751181510352602328834495299, 9.78660989745057170614338105371, 10.45356760984928911950970897721, 11.3633445358452275510955557509, 12.92723805034762503897579693663, 13.75722712304923779845910178705, 14.307436193214537860237136271529, 15.097393483462897314991379171431, 15.64608064716780448948143500679, 16.464501699083739369031691962354, 17.165010659376232449247073758908, 18.16434451991933514660069744958, 19.02048727716872985220582738919, 19.488085221108961275091291331860, 20.29187165176895629375177113109, 21.390502430326006084619980277702, 22.18624508160843458056293912758

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