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Properties

Label 1-1021-1021.31-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.282 + 0.959i$
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  + (−0.557 + 0.830i)2-s + (−0.932 + 0.361i)3-s + (−0.378 − 0.925i)4-s + (−0.297 − 0.954i)5-s + (0.219 − 0.975i)6-s + (0.326 + 0.945i)7-s + (0.979 + 0.201i)8-s + (0.739 − 0.673i)9-s + (0.958 + 0.285i)10-s + (−0.0799 − 0.996i)11-s + (0.687 + 0.726i)12-s + (0.961 − 0.273i)13-s + (−0.966 − 0.255i)14-s + (0.622 + 0.782i)15-s + (−0.713 + 0.700i)16-s + (−0.489 − 0.872i)17-s + ⋯
L(s)  = 1  + (−0.557 + 0.830i)2-s + (−0.932 + 0.361i)3-s + (−0.378 − 0.925i)4-s + (−0.297 − 0.954i)5-s + (0.219 − 0.975i)6-s + (0.326 + 0.945i)7-s + (0.979 + 0.201i)8-s + (0.739 − 0.673i)9-s + (0.958 + 0.285i)10-s + (−0.0799 − 0.996i)11-s + (0.687 + 0.726i)12-s + (0.961 − 0.273i)13-s + (−0.966 − 0.255i)14-s + (0.622 + 0.782i)15-s + (−0.713 + 0.700i)16-s + (−0.489 − 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.282 + 0.959i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.282 + 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8056710724 + 0.6025042502i\)
\(L(\frac12)\) \(\approx\) \(0.8056710724 + 0.6025042502i\)
\(L(1)\) \(\approx\) \(0.6109930112 + 0.2040926344i\)
\(L(1)\) \(\approx\) \(0.6109930112 + 0.2040926344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.557 + 0.830i)T \)
3 \( 1 + (-0.932 + 0.361i)T \)
5 \( 1 + (-0.297 - 0.954i)T \)
7 \( 1 + (0.326 + 0.945i)T \)
11 \( 1 + (-0.0799 - 0.996i)T \)
13 \( 1 + (0.961 - 0.273i)T \)
17 \( 1 + (-0.489 - 0.872i)T \)
19 \( 1 + (0.976 + 0.213i)T \)
23 \( 1 + (0.963 - 0.267i)T \)
29 \( 1 + (-0.755 + 0.655i)T \)
31 \( 1 + (0.505 + 0.862i)T \)
37 \( 1 + (-0.0492 + 0.998i)T \)
41 \( 1 + (-0.213 + 0.976i)T \)
43 \( 1 + (-0.0861 + 0.996i)T \)
47 \( 1 + (0.704 - 0.709i)T \)
53 \( 1 + (0.999 + 0.0431i)T \)
59 \( 1 + (0.934 + 0.355i)T \)
61 \( 1 + (-0.261 - 0.965i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.201 + 0.979i)T \)
73 \( 1 + (-0.966 + 0.255i)T \)
79 \( 1 + (0.478 - 0.878i)T \)
83 \( 1 + (0.153 - 0.988i)T \)
89 \( 1 + (-0.0307 - 0.999i)T \)
97 \( 1 + (-0.920 - 0.389i)T \)
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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

−21.09803384075221762365911721905, −20.508132028077327864330479639720, −19.4619004745866351592775680874, −18.91870094196522339090127844522, −18.04033069752514174346982257605, −17.587865840608873532010047567801, −16.938187451731775762577682333226, −15.94176419680801543125276479391, −15.04134093572731872346941582233, −13.6987761708516162799513740499, −13.285836688804119279631785296535, −12.20490006692768868259355528874, −11.44235224482563501400330314187, −10.8587686516552237686756903946, −10.37121926403062214209270982808, −9.43552834807682199191131379905, −8.11598803927390037227746149374, −7.24154995774945191505132266210, −6.944564157599775799062177915759, −5.597160012339863250951078073719, −4.24570246706215995886588322040, −3.8361578840109992299559511792, −2.38298239817080321018247819043, −1.481567453071603526110437552390, −0.47455241861698730000078672238, 0.73407345660595498218940862208, 1.37115758427307003891223290366, 3.27953914079933174515266909693, 4.645900311426552471275390822749, 5.24058183099382130118214047227, 5.81871301623619464851959050144, 6.72810186352425468487771618480, 7.867949911336180742065453335278, 8.78835072712925369434195522680, 9.12824872946582881277968703647, 10.2437440724980458577205175807, 11.31079748312732942527053872150, 11.66340320814869298923070768555, 12.908744041987174210970908837, 13.632191628547710300621118602947, 14.862171168594882051624875469257, 15.68577687324424885617848130505, 16.12003299827582618954777534156, 16.66511891413822757191611982145, 17.59413482028688238706564267728, 18.37669064440720142757075840476, 18.74826151519462118955855402147, 20.020059957537488542348243549571, 20.813602701597746280618736484137, 21.63626525953388684015847067187

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