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Properties

Label 1-1021-1021.32-r1-0-0
Degree $1$
Conductor $1021$
Sign $-0.564 - 0.825i$
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  + (−0.895 − 0.445i)2-s + (−0.0922 − 0.995i)3-s + (0.602 + 0.798i)4-s + (0.932 + 0.361i)5-s + (−0.361 + 0.932i)6-s + (−0.526 − 0.850i)7-s + (−0.183 − 0.982i)8-s + (−0.982 + 0.183i)9-s + (−0.673 − 0.739i)10-s + (0.739 + 0.673i)11-s + (0.739 − 0.673i)12-s + (0.895 − 0.445i)13-s + (0.0922 + 0.995i)14-s + (0.273 − 0.961i)15-s + (−0.273 + 0.961i)16-s + (−0.932 − 0.361i)17-s + ⋯
L(s)  = 1  + (−0.895 − 0.445i)2-s + (−0.0922 − 0.995i)3-s + (0.602 + 0.798i)4-s + (0.932 + 0.361i)5-s + (−0.361 + 0.932i)6-s + (−0.526 − 0.850i)7-s + (−0.183 − 0.982i)8-s + (−0.982 + 0.183i)9-s + (−0.673 − 0.739i)10-s + (0.739 + 0.673i)11-s + (0.739 − 0.673i)12-s + (0.895 − 0.445i)13-s + (0.0922 + 0.995i)14-s + (0.273 − 0.961i)15-s + (−0.273 + 0.961i)16-s + (−0.932 − 0.361i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.564 - 0.825i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ -0.564 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6731135356 - 1.275415764i\)
\(L(\frac12)\) \(\approx\) \(0.6731135356 - 1.275415764i\)
\(L(1)\) \(\approx\) \(0.7101444500 - 0.4251566849i\)
\(L(1)\) \(\approx\) \(0.7101444500 - 0.4251566849i\)

Euler product

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

−21.56888612089646455998388247610, −21.07428299579101850597658975747, −19.82418932790913693263991866852, −19.542151019158331503310063072499, −18.30869422646218779083090573702, −17.638179958059336192261709815470, −16.99347855999299251832483552565, −16.037305102817662599934176584369, −15.832242396386888950272414767686, −14.80334729969787565813180067283, −13.995355026899416681165017537805, −13.12194778835846143672088762667, −11.66348672394016646323114328403, −11.17801061763122271887886436124, −10.1590891800058123441129283411, −9.426023598791034937499995867793, −8.84042280758937195797269941183, −8.477875797975200147724294967234, −6.64630680188513673617131015888, −6.1471341116210811214342356916, −5.47672584080161605421064959076, −4.41394363966404139001476690180, −3.07244604688842169725743917165, −2.0926913898522371568229905156, −0.873192980208643699715009950320, 0.514648600082756155749424328540, 1.39312286870680866491436602981, 2.21938024678374839471655923816, 3.13957966825781448478081252716, 4.26730635392810499465882969470, 6.08169790341508654674794534440, 6.49024741730906621916939011328, 7.240396902114440420710873568037, 8.153541017385115932044840823934, 9.08804273294357797075737028043, 9.89137577196248130961516958564, 10.70110856787100623852926307706, 11.348931398860606016459340674429, 12.53683934774288733714130690780, 12.966510190141868586019558143202, 13.8337236145041783399658071441, 14.6174080709063115867035109565, 15.99874967273788921423521444298, 16.832553637584064263698480295068, 17.43006292831345823341064571267, 18.06971211918203286805366859165, 18.65002317826507501625346314987, 19.50080198323271749408672091268, 20.38370520947504076437091819709, 20.60184919996949950953777216009

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