↑

Properties

Label 1-1021-1021.12-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.999 + 0.0166i$
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  + (−0.886 − 0.462i)2-s + (0.0922 − 0.995i)3-s + (0.572 + 0.819i)4-s + (−0.542 + 0.840i)5-s + (−0.542 + 0.840i)6-s + (0.997 − 0.0738i)7-s + (−0.128 − 0.991i)8-s + (−0.982 − 0.183i)9-s + (0.869 − 0.494i)10-s + (−0.412 − 0.911i)11-s + (0.869 − 0.494i)12-s + (0.445 − 0.895i)13-s + (−0.918 − 0.395i)14-s + (0.786 + 0.617i)15-s + (−0.343 + 0.938i)16-s + (−0.0554 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.886 − 0.462i)2-s + (0.0922 − 0.995i)3-s + (0.572 + 0.819i)4-s + (−0.542 + 0.840i)5-s + (−0.542 + 0.840i)6-s + (0.997 − 0.0738i)7-s + (−0.128 − 0.991i)8-s + (−0.982 − 0.183i)9-s + (0.869 − 0.494i)10-s + (−0.412 − 0.911i)11-s + (0.869 − 0.494i)12-s + (0.445 − 0.895i)13-s + (−0.918 − 0.395i)14-s + (0.786 + 0.617i)15-s + (−0.343 + 0.938i)16-s + (−0.0554 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.999 + 0.0166i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.999 + 0.0166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005038020521 - 0.6043648133i\)
\(L(\frac12)\) \(\approx\) \(0.005038020521 - 0.6043648133i\)
\(L(1)\) \(\approx\) \(0.5337618851 - 0.3640108928i\)
\(L(1)\) \(\approx\) \(0.5337618851 - 0.3640108928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.886 - 0.462i)T \)
3 \( 1 + (0.0922 - 0.995i)T \)
5 \( 1 + (-0.542 + 0.840i)T \)
7 \( 1 + (0.997 - 0.0738i)T \)
11 \( 1 + (-0.412 - 0.911i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + (-0.0554 - 0.998i)T \)
19 \( 1 + (-0.982 + 0.183i)T \)
23 \( 1 + (-0.343 - 0.938i)T \)
29 \( 1 + (0.997 + 0.0738i)T \)
31 \( 1 + (0.687 + 0.726i)T \)
37 \( 1 + (0.975 - 0.219i)T \)
41 \( 1 + (-0.982 + 0.183i)T \)
43 \( 1 + (0.378 + 0.925i)T \)
47 \( 1 + (-0.918 - 0.395i)T \)
53 \( 1 + (0.830 + 0.557i)T \)
59 \( 1 + (-0.659 + 0.751i)T \)
61 \( 1 + (-0.918 - 0.395i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.128 - 0.991i)T \)
73 \( 1 + (-0.918 + 0.395i)T \)
79 \( 1 + (-0.993 - 0.110i)T \)
83 \( 1 + (0.0922 - 0.995i)T \)
89 \( 1 + (-0.602 - 0.798i)T \)
97 \( 1 + (-0.850 - 0.526i)T \)
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

−21.58295440675650907131334078428, −21.09203339087600794239010260199, −20.380725892811922248147490816879, −19.72231404217938454345633345050, −18.94793996812749640873660846715, −17.78444708993464398982310810345, −17.17692952724176859076829498026, −16.62434387704525476421180245196, −15.583490423283085976943045499949, −15.29846203091736237498737008631, −14.51873901727221406161323811752, −13.485938692364708788518321769692, −12.078047711362255394297842051170, −11.43036235235824020349428335483, −10.64758366885030279684809591194, −9.813426980354314276314958454526, −8.96765484747949852055396871658, −8.30072164107563709751821468375, −7.80415853189504442114445006441, −6.485201773551090937511902260, −5.43144111368212336877606404333, −4.62647715855825308663413430908, −4.03265457245132318264971095700, −2.306880192632144184288940078855, −1.41312937898299695859143051498, 0.35587364314230254782583443322, 1.398128925021239316448745424149, 2.688964558500757133796103362099, 2.99799382718305794231844813685, 4.40711148659089679613772839549, 5.93727340707463158958729516154, 6.73481841951147184064542048682, 7.623793556172104447678489774346, 8.26891862844195727862679014149, 8.63256374593006167404698596580, 10.272351555088852267475908952960, 10.852516927027303343491276916363, 11.5308984153293867028739441769, 12.17373819114137433795984883557, 13.18245039211726193288531401375, 14.03103767546422912034411015045, 14.8827313207001392493045819727, 15.83612819055858626511228494915, 16.73637173369772460783327308248, 17.77434950426493500007471917914, 18.23080527461149679221974026146, 18.66766336294841762392554076777, 19.56108050300249889640114226226, 20.16180001358350005257492209649, 21.02418305755211226987175633488

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