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Properties

Label 1-1021-1021.73-r0-0-0
Degree $1$
Conductor $1021$
Sign $-0.289 - 0.957i$
Analytic cond. $4.74150$
Root an. cond. $4.74150$
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  + (0.956 − 0.291i)2-s + (−0.982 − 0.183i)3-s + (0.830 − 0.557i)4-s + (−0.993 + 0.110i)5-s + (−0.993 + 0.110i)6-s + (0.165 − 0.986i)7-s + (0.631 − 0.775i)8-s + (0.932 + 0.361i)9-s + (−0.918 + 0.395i)10-s + (0.975 − 0.219i)11-s + (−0.918 + 0.395i)12-s + (−0.602 − 0.798i)13-s + (−0.128 − 0.991i)14-s + (0.997 + 0.0738i)15-s + (0.378 − 0.925i)16-s + (0.869 + 0.494i)17-s + ⋯
L(s)  = 1  + (0.956 − 0.291i)2-s + (−0.982 − 0.183i)3-s + (0.830 − 0.557i)4-s + (−0.993 + 0.110i)5-s + (−0.993 + 0.110i)6-s + (0.165 − 0.986i)7-s + (0.631 − 0.775i)8-s + (0.932 + 0.361i)9-s + (−0.918 + 0.395i)10-s + (0.975 − 0.219i)11-s + (−0.918 + 0.395i)12-s + (−0.602 − 0.798i)13-s + (−0.128 − 0.991i)14-s + (0.997 + 0.0738i)15-s + (0.378 − 0.925i)16-s + (0.869 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.088057247 - 1.466059770i\)
\(L(\frac12)\) \(\approx\) \(1.088057247 - 1.466059770i\)
\(L(1)\) \(\approx\) \(1.196371089 - 0.6159847706i\)
\(L(1)\) \(\approx\) \(1.196371089 - 0.6159847706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.956 - 0.291i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.993 + 0.110i)T \)
7 \( 1 + (0.165 - 0.986i)T \)
11 \( 1 + (0.975 - 0.219i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (0.869 + 0.494i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.378 + 0.925i)T \)
29 \( 1 + (0.165 + 0.986i)T \)
31 \( 1 + (-0.966 + 0.255i)T \)
37 \( 1 + (-0.478 + 0.878i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (-0.886 - 0.462i)T \)
47 \( 1 + (-0.128 - 0.991i)T \)
53 \( 1 + (0.237 - 0.971i)T \)
59 \( 1 + (0.903 - 0.429i)T \)
61 \( 1 + (-0.128 - 0.991i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.631 - 0.775i)T \)
73 \( 1 + (-0.128 + 0.991i)T \)
79 \( 1 + (0.510 - 0.859i)T \)
83 \( 1 + (-0.982 - 0.183i)T \)
89 \( 1 + (-0.273 + 0.961i)T \)
97 \( 1 + (0.445 + 0.895i)T \)
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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

−22.07175387761594299342927578766, −21.27792615446599403632672586698, −20.57766920952708799302264724556, −19.50243815827470073989293047370, −18.75692093388870141129817908310, −17.79165168748610208242327794326, −16.64438727775216267614200988493, −16.41830752933979129376809134697, −15.5487273852175847451131030171, −14.7709352511447633948763319045, −14.22440103639871927510841023356, −12.70278057824088548985926724910, −12.24074438046009348619914704135, −11.62989790372680587243556028050, −11.24219462537149311739132592033, −9.85098301633369284002780442220, −8.8801160905142140339554593525, −7.640609048361320892073632891304, −7.03477092986609377251994895867, −6.08540782226461423667122991842, −5.2915843061357279967927506964, −4.49687908397113566296415766962, −3.832792632385169749560530737094, −2.6568064118577384263232056915, −1.30380469127636469623440257035, 0.73808678044098084917927486973, 1.54869629714540517274016064402, 3.39322739283626395480469414703, 3.7268683846204014313994518908, 4.92424511386299000945399607545, 5.399477580390865025122837739, 6.737948516972915442913332752703, 7.1598983777211636873774240436, 7.96792897667047572364795526489, 9.692954585680646977761161082196, 10.53767592705307410575626027149, 11.1810957572685708201083096343, 11.8484178758645419154502355927, 12.46752285730645967488459884488, 13.29242883558185967434061882894, 14.25966677865860369574678070529, 14.9808529183104309687829833994, 15.87474721086943865211910872253, 16.6042311082479981764767852311, 17.222249731015106227804367261036, 18.319813546865300695843587937527, 19.37456393909712826288644901248, 19.79755660841402805947013809392, 20.559717439303832977235277929313, 21.70730106020657876121737684428

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