↑

Properties

Label 1-1021-1021.866-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.419 - 0.907i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (−0.961 + 0.273i)2-s + (−0.932 − 0.361i)3-s + (0.850 − 0.526i)4-s + (0.0922 − 0.995i)5-s + (0.995 + 0.0922i)6-s + (−0.798 − 0.602i)7-s + (−0.673 + 0.739i)8-s + (0.739 + 0.673i)9-s + (0.183 + 0.982i)10-s + (−0.982 − 0.183i)11-s + (−0.982 + 0.183i)12-s + (0.961 + 0.273i)13-s + (0.932 + 0.361i)14-s + (−0.445 + 0.895i)15-s + (0.445 − 0.895i)16-s + (−0.0922 + 0.995i)17-s + ⋯
L(s)  = 1  + (−0.961 + 0.273i)2-s + (−0.932 − 0.361i)3-s + (0.850 − 0.526i)4-s + (0.0922 − 0.995i)5-s + (0.995 + 0.0922i)6-s + (−0.798 − 0.602i)7-s + (−0.673 + 0.739i)8-s + (0.739 + 0.673i)9-s + (0.183 + 0.982i)10-s + (−0.982 − 0.183i)11-s + (−0.982 + 0.183i)12-s + (0.961 + 0.273i)13-s + (0.932 + 0.361i)14-s + (−0.445 + 0.895i)15-s + (0.445 − 0.895i)16-s + (−0.0922 + 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.419 - 0.907i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.419 - 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2627411660 - 0.4109117320i\)
\(L(\frac12)\) \(\approx\) \(0.2627411660 - 0.4109117320i\)
\(L(1)\) \(\approx\) \(0.4481699110 - 0.1103851929i\)
\(L(1)\) \(\approx\) \(0.4481699110 - 0.1103851929i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.961 + 0.273i)T \)
3 \( 1 + (-0.932 - 0.361i)T \)
5 \( 1 + (0.0922 - 0.995i)T \)
7 \( 1 + (-0.798 - 0.602i)T \)
11 \( 1 + (-0.982 - 0.183i)T \)
13 \( 1 + (0.961 + 0.273i)T \)
17 \( 1 + (-0.0922 + 0.995i)T \)
19 \( 1 + (-0.673 - 0.739i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (0.602 + 0.798i)T \)
31 \( 1 + (-0.673 + 0.739i)T \)
37 \( 1 + (0.361 - 0.932i)T \)
41 \( 1 + (-0.739 + 0.673i)T \)
43 \( 1 + (-0.798 - 0.602i)T \)
47 \( 1 + (0.932 + 0.361i)T \)
53 \( 1 + (0.895 - 0.445i)T \)
59 \( 1 + (0.361 - 0.932i)T \)
61 \( 1 + (-0.932 - 0.361i)T \)
67 \( 1 + T \)
71 \( 1 + (0.739 + 0.673i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (0.982 - 0.183i)T \)
83 \( 1 + (-0.932 - 0.361i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (0.798 + 0.602i)T \)
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

−21.48045797151602352131830273495, −21.01402306066016746287797302598, −20.0945260766402041432705410645, −18.761467343925135503130818063874, −18.53132700762017933639808297862, −18.08492706113734253929602066961, −16.94017070521633737835982084191, −16.33407995511493478524807984799, −15.36498384945630658708679730743, −15.24887144911479085512972757082, −13.5160726063181263759919021888, −12.64608454291792027474616285908, −11.83226800957189231292065175249, −11.070515333712974617756428910046, −10.335171403319502842317883212525, −9.92750850248056218671964631091, −8.911141707813333963520948066398, −7.8646765524512142284739521040, −6.819684987830370207110229934521, −6.286215237627184165854465244, −5.47105272475275636681944885729, −3.95531016145608167892665366163, −2.99043244012018512706020307773, −2.209991605560205131568940641175, −0.64668098288348781324754811524, 0.28378325429928291737328837056, 1.09853472029821043167346098173, 2.0158254778544848311797445777, 3.584578033354647506808051423403, 4.902707652844429857255075899093, 5.7017579550722700443420757156, 6.48249644174296839509074017160, 7.23700893072562548413812762954, 8.22850902546631734641410611350, 8.93967960085515787419627000484, 9.96983650033814419196227399369, 10.71386894636688419043867998644, 11.26553315722663475643279216158, 12.50736739971204873824705066555, 12.99969594819899933467969552134, 13.80930755759308089030944842251, 15.463231231545173573027540523873, 15.911415064042217586509785458846, 16.65427849756504659342359854798, 17.13826276834099894982161522872, 17.93239267045985990001006585306, 18.67301122608427505425877693934, 19.55252939179672605937716219077, 20.05436103349271243079992902929, 21.224717062003847622916794048541

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