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Properties

Label 1-1021-1021.288-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.541 - 0.840i$
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  + (0.673 − 0.739i)2-s + (−0.445 − 0.895i)3-s + (−0.0922 − 0.995i)4-s + (−0.273 + 0.961i)5-s + (−0.961 − 0.273i)6-s + (−0.361 + 0.932i)7-s + (−0.798 − 0.602i)8-s + (−0.602 + 0.798i)9-s + (0.526 + 0.850i)10-s + (−0.850 − 0.526i)11-s + (−0.850 + 0.526i)12-s + (−0.673 − 0.739i)13-s + (0.445 + 0.895i)14-s + (0.982 − 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.273 − 0.961i)17-s + ⋯
L(s)  = 1  + (0.673 − 0.739i)2-s + (−0.445 − 0.895i)3-s + (−0.0922 − 0.995i)4-s + (−0.273 + 0.961i)5-s + (−0.961 − 0.273i)6-s + (−0.361 + 0.932i)7-s + (−0.798 − 0.602i)8-s + (−0.602 + 0.798i)9-s + (0.526 + 0.850i)10-s + (−0.850 − 0.526i)11-s + (−0.850 + 0.526i)12-s + (−0.673 − 0.739i)13-s + (0.445 + 0.895i)14-s + (0.982 − 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.273 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.541 - 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7248222808 - 0.3951091834i\)
\(L(\frac12)\) \(\approx\) \(0.7248222808 - 0.3951091834i\)
\(L(1)\) \(\approx\) \(0.7039707494 - 0.4384897544i\)
\(L(1)\) \(\approx\) \(0.7039707494 - 0.4384897544i\)

Euler product

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

−21.61190892055783799867800075090, −20.93168820963607367564611461789, −20.266298760829222996351903936894, −19.48879698307870537639889535570, −17.96933622696435111778931165120, −17.08863839938553901330647708041, −16.77992623178756331400934050638, −16.08182033107291056436295617265, −15.32804843105759816145593134673, −14.66043938610340162572360760133, −13.613371364345195041543112089456, −12.813462016579661283327177461551, −12.20885724270729592241596228242, −11.24868082914568804833095210053, −10.241708520650740833203995665887, −9.410730582360627878512250367319, −8.501378360911070451259309310745, −7.594940934553744163745428382598, −6.72120617712269573530908011479, −5.65469761991493232248170772582, −4.967085159927873206727682980166, −4.095223020768332296158075597445, −3.78763462699259973242797495263, −2.195120352498982285268421408011, −0.299461817359139613940976217306, 0.440406585107571417025369925531, 2.07111007354643415235990906293, 2.62272665762116817850344246544, 3.368767448325289116562361492295, 4.808031254610532918873109713577, 5.89697117555521360383996131994, 6.04852995809473353645148201028, 7.36189576556010584331508065910, 8.07049503958170993911738447388, 9.45317776713927769749906325131, 10.34623727190402739588061788170, 11.14881048167037423007074626779, 11.70747017840733108415008228818, 12.680002747373375150926491083935, 12.97066938797059374294349541407, 14.16295137642787268864409288942, 14.68952510054738289016789461698, 15.62264403866154878528617608951, 16.44267434686256794123115719635, 17.90749307000443743935852009128, 18.44977530492741419319125937057, 18.824310484453375142226449133612, 19.6159434420827788909132558564, 20.42252429120762285037851177544, 21.628354922545246381870802142932

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