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Properties

Label 1-1021-1021.303-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

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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: $\chi_{1021} (303, \cdot )$
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 \( 1 + (0.798 - 0.602i)T \)
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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

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

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