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Properties

Label 1-4021-4021.98-r1-0-0
Degree $1$
Conductor $4021$
Sign $-0.222 - 0.974i$
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  + (−0.310 + 0.950i)2-s + (0.934 − 0.354i)3-s + (−0.807 − 0.590i)4-s + (−0.483 + 0.875i)5-s + (0.0468 + 0.998i)6-s + (0.951 + 0.309i)7-s + (0.811 − 0.583i)8-s + (0.748 − 0.663i)9-s + (−0.681 − 0.731i)10-s + (−0.0874 − 0.996i)11-s + (−0.964 − 0.265i)12-s + (0.116 − 0.993i)13-s + (−0.589 + 0.808i)14-s + (−0.141 + 0.989i)15-s + (0.303 + 0.952i)16-s + (−0.560 − 0.827i)17-s + ⋯
L(s)  = 1  + (−0.310 + 0.950i)2-s + (0.934 − 0.354i)3-s + (−0.807 − 0.590i)4-s + (−0.483 + 0.875i)5-s + (0.0468 + 0.998i)6-s + (0.951 + 0.309i)7-s + (0.811 − 0.583i)8-s + (0.748 − 0.663i)9-s + (−0.681 − 0.731i)10-s + (−0.0874 − 0.996i)11-s + (−0.964 − 0.265i)12-s + (0.116 − 0.993i)13-s + (−0.589 + 0.808i)14-s + (−0.141 + 0.989i)15-s + (0.303 + 0.952i)16-s + (−0.560 − 0.827i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.222 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4021\)
Sign: $-0.222 - 0.974i$
Analytic conductor: \(432.116\)
Root analytic conductor: \(432.116\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4021} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4021,\ (1:\ ),\ -0.222 - 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8519192055 - 1.068138414i\)
\(L(\frac12)\) \(\approx\) \(0.8519192055 - 1.068138414i\)
\(L(1)\) \(\approx\) \(1.079733597 + 0.1913324833i\)
\(L(1)\) \(\approx\) \(1.079733597 + 0.1913324833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4021 \( 1 \)
good2 \( 1 + (-0.310 + 0.950i)T \)
3 \( 1 + (0.934 - 0.354i)T \)
5 \( 1 + (-0.483 + 0.875i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.0874 - 0.996i)T \)
13 \( 1 + (0.116 - 0.993i)T \)
17 \( 1 + (-0.560 - 0.827i)T \)
19 \( 1 + (-0.649 + 0.760i)T \)
23 \( 1 + (0.769 - 0.638i)T \)
29 \( 1 + (-0.124 - 0.992i)T \)
31 \( 1 + (-0.995 + 0.0920i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
41 \( 1 + (0.965 + 0.261i)T \)
43 \( 1 + (0.775 + 0.631i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.854 + 0.520i)T \)
59 \( 1 + (0.689 + 0.724i)T \)
61 \( 1 + (-0.596 + 0.802i)T \)
67 \( 1 + (-0.698 - 0.715i)T \)
71 \( 1 + (0.0889 - 0.996i)T \)
73 \( 1 + (-0.447 - 0.894i)T \)
79 \( 1 + (-0.328 + 0.944i)T \)
83 \( 1 + (0.844 + 0.536i)T \)
89 \( 1 + (-0.996 + 0.0874i)T \)
97 \( 1 + (0.276 - 0.961i)T \)
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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

−18.80818022427239029409700441294, −17.77378349758005555171536658215, −17.30062718649403898946250908086, −16.58587054242970786077550380153, −15.7668150067503772872222839447, −14.91352103286450482480592104544, −14.46156572386483105394229198285, −13.508891205178013608999157405892, −12.97435014276077535528441142759, −12.44376107281389932800488803594, −11.4103236475461373002667152962, −11.02543588386859254151181276369, −10.1864459710420545378832681697, −9.315607407068705368270938458914, −8.853238478038142724593631931298, −8.392433565530955375557450534554, −7.515847897523292888689974232168, −7.04037632477653447472664351158, −5.22357298089339076596760165885, −4.6303417715361924035667064722, −4.185536277654916819258659501641, −3.56059074456985068454169156008, −2.35585985781580423537389981029, −1.745357082162269713420850981883, −1.201726620338163260228180293200, 0.20496109584516977948217871998, 1.01738977313169169726836616996, 2.184496406254161333042109969567, 2.91680391332011713884984450580, 3.84970608714468553927927727094, 4.52110503246804965653601612626, 5.6400373151325682004973164056, 6.20342955718829658753734076787, 7.161608683635683162571048846791, 7.666781997071336887314716614751, 8.22337925284067531067289378690, 8.77348087792758016122024675741, 9.4978655345536861735228017530, 10.63978731845021481062391065335, 10.88379862171425092017069747981, 11.98325843275582706776311734348, 12.93199434316286592467368473055, 13.5966244010283619041450040978, 14.27955138387559825196265188592, 14.80726244659822655011718276000, 15.17507985267224321637559273352, 15.91986832353510279380368364972, 16.6235226908865493066834101034, 17.718701375732626829069356768047, 18.24050721085597164335314612101

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