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Properties

Label 1-1021-1021.163-r0-0-0
Degree $1$
Conductor $1021$
Sign $0.948 + 0.316i$
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  + (−0.903 − 0.429i)2-s + (−0.273 − 0.961i)3-s + (0.631 + 0.775i)4-s + (0.771 + 0.636i)5-s + (−0.165 + 0.986i)6-s + (−0.510 + 0.859i)7-s + (−0.237 − 0.971i)8-s + (−0.850 + 0.526i)9-s + (−0.423 − 0.905i)10-s + (0.755 − 0.655i)11-s + (0.572 − 0.819i)12-s + (0.982 + 0.183i)13-s + (0.830 − 0.557i)14-s + (0.401 − 0.916i)15-s + (−0.201 + 0.979i)16-s + (0.963 + 0.267i)17-s + ⋯
L(s)  = 1  + (−0.903 − 0.429i)2-s + (−0.273 − 0.961i)3-s + (0.631 + 0.775i)4-s + (0.771 + 0.636i)5-s + (−0.165 + 0.986i)6-s + (−0.510 + 0.859i)7-s + (−0.237 − 0.971i)8-s + (−0.850 + 0.526i)9-s + (−0.423 − 0.905i)10-s + (0.755 − 0.655i)11-s + (0.572 − 0.819i)12-s + (0.982 + 0.183i)13-s + (0.830 − 0.557i)14-s + (0.401 − 0.916i)15-s + (−0.201 + 0.979i)16-s + (0.963 + 0.267i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(4.74150\)
Root analytic conductor: \(4.74150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (0:\ ),\ 0.948 + 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9290451168 + 0.1510792309i\)
\(L(\frac12)\) \(\approx\) \(0.9290451168 + 0.1510792309i\)
\(L(1)\) \(\approx\) \(0.7500656991 - 0.09839601042i\)
\(L(1)\) \(\approx\) \(0.7500656991 - 0.09839601042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (-0.903 - 0.429i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (0.771 + 0.636i)T \)
7 \( 1 + (-0.510 + 0.859i)T \)
11 \( 1 + (0.755 - 0.655i)T \)
13 \( 1 + (0.982 + 0.183i)T \)
17 \( 1 + (0.963 + 0.267i)T \)
19 \( 1 + (-0.881 + 0.473i)T \)
23 \( 1 + (-0.747 + 0.664i)T \)
29 \( 1 + (0.489 - 0.872i)T \)
31 \( 1 + (0.990 - 0.135i)T \)
37 \( 1 + (-0.531 + 0.846i)T \)
41 \( 1 + (0.881 - 0.473i)T \)
43 \( 1 + (-0.980 + 0.195i)T \)
47 \( 1 + (-0.897 - 0.440i)T \)
53 \( 1 + (-0.995 + 0.0984i)T \)
59 \( 1 + (0.927 - 0.372i)T \)
61 \( 1 + (0.0677 + 0.997i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (0.237 + 0.971i)T \)
73 \( 1 + (0.830 + 0.557i)T \)
79 \( 1 + (0.0184 + 0.999i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (-0.153 - 0.988i)T \)
97 \( 1 + (0.908 + 0.417i)T \)
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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

−21.24789666007518712874400392183, −20.72762174076801446910823830414, −20.03401696628035078761495048153, −19.38855766485485292610232405756, −18.028194260516588428359869214663, −17.541403989507860560726371361726, −16.74190024400316405195615884833, −16.34456131475149996712928344853, −15.631341368530253163510407070519, −14.49655283546026861712538133445, −14.02783708045668215238937210146, −12.774320946786132229410648683727, −11.799105329790179213684333023379, −10.67139949755186903695805009095, −10.22399676226576143538867502039, −9.49545022133138782533340543828, −8.85871873003646814480100197732, −7.98020422373877693654318080189, −6.54616341927009310543720040350, −6.25537228221180729021900991180, −5.095558725973873983808668586186, −4.29513736615134043087892180140, −3.102396029652449540153437982539, −1.66193882299747276247679601146, −0.63052592246910810293237207676, 1.16812199771235115305671053834, 1.93072665839293250391523474611, 2.85117904533835069364994768042, 3.67143318104627735699534330344, 5.756091176970011240373800483862, 6.23650142058337653080175184600, 6.79009521078143408891664957259, 8.1179154735636653773982290888, 8.571090176334020503431630383569, 9.64316411614596991306503782284, 10.342446486402659124616903444543, 11.42281361867474396655257944818, 11.83982392961431013112410244733, 12.79171915967414035465989924393, 13.546806922220254401908924272025, 14.383793112176702254743858286280, 15.53939094593847273965356359942, 16.51274393378346076354906790398, 17.18362471619543181343241107885, 17.90105534733220363571767785394, 18.686869494924073642426315427895, 19.035008472731010465936400526762, 19.62383419831531982975308001845, 21.00209308207627337293106361318, 21.465428589736187019482816773363

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