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Properties

Label 1-1021-1021.909-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.999 + 0.0256i$
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  + (−0.361 − 0.932i)2-s + (0.850 − 0.526i)3-s + (−0.739 + 0.673i)4-s + (−0.602 + 0.798i)5-s + (−0.798 − 0.602i)6-s + (−0.183 − 0.982i)7-s + (0.895 + 0.445i)8-s + (0.445 − 0.895i)9-s + (0.961 + 0.273i)10-s + (−0.273 − 0.961i)11-s + (−0.273 + 0.961i)12-s + (0.361 − 0.932i)13-s + (−0.850 + 0.526i)14-s + (−0.0922 + 0.995i)15-s + (0.0922 − 0.995i)16-s + (0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (−0.361 − 0.932i)2-s + (0.850 − 0.526i)3-s + (−0.739 + 0.673i)4-s + (−0.602 + 0.798i)5-s + (−0.798 − 0.602i)6-s + (−0.183 − 0.982i)7-s + (0.895 + 0.445i)8-s + (0.445 − 0.895i)9-s + (0.961 + 0.273i)10-s + (−0.273 − 0.961i)11-s + (−0.273 + 0.961i)12-s + (0.361 − 0.932i)13-s + (−0.850 + 0.526i)14-s + (−0.0922 + 0.995i)15-s + (0.0922 − 0.995i)16-s + (0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.999 + 0.0256i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.999 + 0.0256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02606118788 - 2.034768816i\)
\(L(\frac12)\) \(\approx\) \(0.02606118788 - 2.034768816i\)
\(L(1)\) \(\approx\) \(0.7635873283 - 0.7845942207i\)
\(L(1)\) \(\approx\) \(0.7635873283 - 0.7845942207i\)

Euler product

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

−21.694985400524734187607516183761, −21.01527360418583048741278448204, −20.11318229189319350289199394621, −19.3566247636432660247691906844, −18.75580818977107138889571260574, −17.94155825811621876921168057399, −16.713114678550899971806824981456, −16.20984107298938677772949013239, −15.58135930333848740676792693840, −14.88508499505288480416457367664, −14.28030399806075624022181522152, −13.1685832840882095961777307857, −12.49548199327558537335140589878, −11.43581228173242663033745766003, −9.93660101106512373739212248033, −9.68716913445449638641512888042, −8.50412729334133865376410536183, −8.37245153861090058237937839349, −7.37947364625269908716491884755, −6.289749730271386260156950068537, −5.17676918731171316922714949495, −4.53828911016225574229860235656, −3.70000675014556560057741955197, −2.284051276393303451813206384171, −1.17538436071557729333904922473, 0.55041308264860046359310598605, 1.066305096865614620610246640128, 2.64121000736412553233963311283, 3.30231128465071562084085597164, 3.65198803445437497227785206620, 5.05151249052629296838713721283, 6.59213332654935905093913106199, 7.535762330324119345859693068719, 7.93382593462846002801796003432, 8.83718477739149985239339045334, 9.96729572059675590372659390311, 10.455715232788235216204246315778, 11.47580389346488725400779377463, 12.06207380805688926877012780073, 13.32411310804531851912398893480, 13.647679186186781139071463737059, 14.32469523511592229802523297947, 15.539540641318146810941176839278, 16.25565278679530473671654250370, 17.5640477488567667807160784515, 18.10148838990861166620728284226, 18.937749715299198410546408768337, 19.42327246501253375333439717123, 20.14442856718753784568796555832, 20.69136448995645544368621639784

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