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Properties

Label 1-1021-1021.28-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.565 + 0.824i$
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  + (0.0738 − 0.997i)2-s + (−0.739 − 0.673i)3-s + (−0.989 − 0.147i)4-s + (0.687 + 0.726i)5-s + (−0.726 + 0.687i)6-s + (−0.938 − 0.343i)7-s + (−0.219 + 0.975i)8-s + (0.0922 + 0.995i)9-s + (0.775 − 0.631i)10-s + (−0.0554 + 0.998i)11-s + (0.631 + 0.775i)12-s + (−0.526 + 0.850i)13-s + (−0.412 + 0.911i)14-s + (−0.0184 − 0.999i)15-s + (0.956 + 0.291i)16-s + (0.128 + 0.991i)17-s + ⋯
L(s)  = 1  + (0.0738 − 0.997i)2-s + (−0.739 − 0.673i)3-s + (−0.989 − 0.147i)4-s + (0.687 + 0.726i)5-s + (−0.726 + 0.687i)6-s + (−0.938 − 0.343i)7-s + (−0.219 + 0.975i)8-s + (0.0922 + 0.995i)9-s + (0.775 − 0.631i)10-s + (−0.0554 + 0.998i)11-s + (0.631 + 0.775i)12-s + (−0.526 + 0.850i)13-s + (−0.412 + 0.911i)14-s + (−0.0184 − 0.999i)15-s + (0.956 + 0.291i)16-s + (0.128 + 0.991i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.565 + 0.824i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.565 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2005150625 + 0.3804050202i\)
\(L(\frac12)\) \(\approx\) \(0.2005150625 + 0.3804050202i\)
\(L(1)\) \(\approx\) \(0.6565826168 - 0.1915875266i\)
\(L(1)\) \(\approx\) \(0.6565826168 - 0.1915875266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.0738 - 0.997i)T \)
3 \( 1 + (-0.739 - 0.673i)T \)
5 \( 1 + (0.687 + 0.726i)T \)
7 \( 1 + (-0.938 - 0.343i)T \)
11 \( 1 + (-0.0554 + 0.998i)T \)
13 \( 1 + (-0.526 + 0.850i)T \)
17 \( 1 + (0.128 + 0.991i)T \)
19 \( 1 + (-0.995 - 0.0922i)T \)
23 \( 1 + (0.956 - 0.291i)T \)
29 \( 1 + (0.343 + 0.938i)T \)
31 \( 1 + (0.751 - 0.659i)T \)
37 \( 1 + (-0.494 - 0.869i)T \)
41 \( 1 + (-0.0922 + 0.995i)T \)
43 \( 1 + (-0.617 + 0.786i)T \)
47 \( 1 + (-0.412 + 0.911i)T \)
53 \( 1 + (-0.326 + 0.945i)T \)
59 \( 1 + (-0.110 + 0.993i)T \)
61 \( 1 + (0.412 - 0.911i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.975 + 0.219i)T \)
73 \( 1 + (-0.412 - 0.911i)T \)
79 \( 1 + (0.966 + 0.255i)T \)
83 \( 1 + (-0.739 - 0.673i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.961 + 0.273i)T \)
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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

−21.40483568886464855802762100083, −20.68984418908151857187704937735, −19.37525180609184830309072801477, −18.60258229566702742686067440466, −17.58496651863257159407307855739, −17.063349998611754141835686874092, −16.46468355104007166498497558926, −15.73173742079013695988119617313, −15.21279455506222935274055964643, −14.03526735691097456233698058816, −13.23759913874466660105515903061, −12.58469141808311116009847693219, −11.72672914251997893049448264664, −10.273868062692898920259681659594, −9.86540010588883079514311302697, −8.93321211748483336380197436165, −8.37688399509024530314336511858, −6.8641665552847276180419695884, −6.268716517205470666933201150803, −5.325679231284063617706152808756, −5.08422941628125799279972777725, −3.80368573244524460519601489173, −2.84457613135088413160509456513, −0.772591598090719097912816425128, −0.138090552614514837558657932840, 1.309582564926821139081347725335, 2.12468374984357871720208068189, 2.92998040201954464497121460778, 4.21158085004063954811021204927, 5.04700291804454820740672909657, 6.28614956152981660186655577809, 6.6820991220365315917490988035, 7.75699992915408749695150632527, 9.1181801543236931820444996268, 9.90805139908650723960503515945, 10.56375160502422409930970549, 11.17191035749709143903278403647, 12.34737586954449332710395660832, 12.74826487757895855340156592120, 13.451256682501737287446362910129, 14.3407736943845786082142734768, 15.11242871425718713927043245901, 16.669365685769723724740939278708, 17.198126641465703096587605536393, 17.85577434885784820701747523219, 18.74592763018333252674036014492, 19.25490307735172566171351263600, 19.8529045324828808061701277112, 21.13332477261715742436807468699, 21.71876705213770691451446403150

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