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Properties

Label 1-4021-4021.57-r1-0-0
Degree $1$
Conductor $4021$
Sign $0.414 - 0.910i$
Analytic cond. $432.116$
Root an. cond. $432.116$
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  + (0.337 − 0.941i)2-s + (−0.329 + 0.944i)3-s + (−0.772 − 0.634i)4-s + (−0.444 − 0.895i)5-s + (0.777 + 0.628i)6-s + i·7-s + (−0.858 + 0.513i)8-s + (−0.782 − 0.622i)9-s + (−0.993 + 0.116i)10-s + (−0.395 − 0.918i)11-s + (0.854 − 0.520i)12-s + (−0.792 − 0.610i)13-s + (0.941 + 0.337i)14-s + (0.992 − 0.124i)15-s + (0.194 + 0.980i)16-s + (−0.591 − 0.806i)17-s + ⋯
L(s)  = 1  + (0.337 − 0.941i)2-s + (−0.329 + 0.944i)3-s + (−0.772 − 0.634i)4-s + (−0.444 − 0.895i)5-s + (0.777 + 0.628i)6-s + i·7-s + (−0.858 + 0.513i)8-s + (−0.782 − 0.622i)9-s + (−0.993 + 0.116i)10-s + (−0.395 − 0.918i)11-s + (0.854 − 0.520i)12-s + (−0.792 − 0.610i)13-s + (0.941 + 0.337i)14-s + (0.992 − 0.124i)15-s + (0.194 + 0.980i)16-s + (−0.591 − 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(432.116\)
Root analytic conductor: \(432.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (1:\ ),\ 0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5117798377 - 0.3294122695i\)
\(L(\frac12)\) \(\approx\) \(0.5117798377 - 0.3294122695i\)
\(L(1)\) \(\approx\) \(0.6530736693 - 0.2622983263i\)
\(L(1)\) \(\approx\) \(0.6530736693 - 0.2622983263i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (0.337 - 0.941i)T \)
3 \( 1 + (-0.329 + 0.944i)T \)
5 \( 1 + (-0.444 - 0.895i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.395 - 0.918i)T \)
13 \( 1 + (-0.792 - 0.610i)T \)
17 \( 1 + (-0.591 - 0.806i)T \)
19 \( 1 + (0.996 - 0.0858i)T \)
23 \( 1 + (0.999 + 0.0390i)T \)
29 \( 1 + (-0.693 + 0.720i)T \)
31 \( 1 + (-0.991 - 0.132i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.999 + 0.00781i)T \)
43 \( 1 + (0.506 + 0.862i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.566 - 0.824i)T \)
59 \( 1 + (0.578 - 0.815i)T \)
61 \( 1 + (0.742 - 0.670i)T \)
67 \( 1 + (0.845 - 0.533i)T \)
71 \( 1 + (0.731 + 0.681i)T \)
73 \( 1 + (0.0234 + 0.999i)T \)
79 \( 1 + (-0.970 + 0.239i)T \)
83 \( 1 + (-0.373 + 0.927i)T \)
89 \( 1 + (-0.918 + 0.395i)T \)
97 \( 1 + (-0.285 + 0.958i)T \)
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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

−18.2983399737737421757958253097, −17.53475969679627847200942782415, −17.24250885606821389051006600981, −16.471776992690197261837856869041, −15.675702579421227073324780745215, −14.84709458214047342252597025875, −14.43003060479568781092111345201, −13.70953451240272231522444606591, −13.07868210660066971234888515074, −12.45193870351626180584359824851, −11.6963747015373165674549509774, −10.97182949766844967054271839828, −10.17598088696309703866700400054, −9.308534009668502930422849861109, −8.29311158197027975471248469480, −7.42518321498474143624030730343, −7.247547064941116711734486013982, −6.80446189813971689774321249447, −5.882436302606374095741168661975, −5.05119690066866372044445664238, −4.27351194991408985481816135595, −3.50804286019641552669341019561, −2.56415952863368427701353310874, −1.623628523078531144576071288469, −0.27928292400621463728778015659, 0.274979608338984294266325283485, 1.20523211668218965248545322918, 2.45466836878951576005586989193, 3.185649198350357216492138453909, 3.68756301424519103258874995761, 4.95158015522278821067604538321, 5.16145487952807730746624225338, 5.51798769051590252067822146099, 6.765175040798042154377113889502, 8.164077633382754645843469875781, 8.63005612019536623708432040032, 9.536884003762206441265571291145, 9.607778360729116896188775482089, 10.8490408053499850875077932512, 11.43705033398963201905341267639, 11.73642504560015305868310218972, 12.7236343485246265968173519802, 13.04034518855974782733930455471, 14.1169976505925491512387400353, 14.81827639197369356935868598298, 15.5378049230711583479555861417, 15.9549295343108856085550913902, 16.78182398576288445960559049099, 17.53746623189972245657042820167, 18.32150179453610160734758775118

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