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Properties

Label 1-421-421.190-r0-0-0
Degree $1$
Conductor $421$
Sign $0.285 + 0.958i$
Analytic cond. $1.95511$
Root an. cond. $1.95511$
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  + (0.134 + 0.990i)2-s + (−0.550 + 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.963 − 0.266i)5-s + (−0.900 − 0.433i)6-s + (−0.550 − 0.834i)7-s + (−0.393 − 0.919i)8-s + (−0.393 − 0.919i)9-s + (0.134 − 0.990i)10-s + (0.936 − 0.351i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (0.753 − 0.657i)14-s + (0.753 − 0.657i)15-s + (0.858 − 0.512i)16-s + (−0.691 + 0.722i)17-s + ⋯
L(s)  = 1  + (0.134 + 0.990i)2-s + (−0.550 + 0.834i)3-s + (−0.963 + 0.266i)4-s + (−0.963 − 0.266i)5-s + (−0.900 − 0.433i)6-s + (−0.550 − 0.834i)7-s + (−0.393 − 0.919i)8-s + (−0.393 − 0.919i)9-s + (0.134 − 0.990i)10-s + (0.936 − 0.351i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (0.753 − 0.657i)14-s + (0.753 − 0.657i)15-s + (0.858 − 0.512i)16-s + (−0.691 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.285 + 0.958i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.285 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5307382664 + 0.3956785496i\)
\(L(\frac12)\) \(\approx\) \(0.5307382664 + 0.3956785496i\)
\(L(1)\) \(\approx\) \(0.5645308400 + 0.3667716323i\)
\(L(1)\) \(\approx\) \(0.5645308400 + 0.3667716323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.134 + 0.990i)T \)
3 \( 1 + (-0.550 + 0.834i)T \)
5 \( 1 + (-0.963 - 0.266i)T \)
7 \( 1 + (-0.550 - 0.834i)T \)
11 \( 1 + (0.936 - 0.351i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.691 + 0.722i)T \)
19 \( 1 + (0.473 + 0.880i)T \)
23 \( 1 + (-0.0448 - 0.998i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.995 - 0.0896i)T \)
37 \( 1 + (0.936 - 0.351i)T \)
41 \( 1 + (0.473 + 0.880i)T \)
43 \( 1 + (0.858 - 0.512i)T \)
47 \( 1 + (-0.963 - 0.266i)T \)
53 \( 1 + (-0.550 - 0.834i)T \)
59 \( 1 + (0.983 + 0.178i)T \)
61 \( 1 + (0.858 - 0.512i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.858 + 0.512i)T \)
83 \( 1 + (0.983 + 0.178i)T \)
89 \( 1 + (-0.995 + 0.0896i)T \)
97 \( 1 + (0.473 - 0.880i)T \)
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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

−23.850398064613072609599873018914, −22.90507611694831431314265460949, −22.29889938021650933932138181558, −21.88391415281783765373039119893, −20.16609844053608925536154006235, −19.594387236444406185148118337085, −19.13260399049889473725509784567, −18.022388263792345459459644891778, −17.5812749914242312211786371186, −16.17562371449949096803567416473, −15.153861791042657489990783790105, −14.158840684204191869023101199, −13.02072901233580634974555075228, −12.338123545246539907516788921163, −11.65578898374028193884777548856, −11.105044157632923962952779207602, −9.73123932908701986297175690338, −8.844973002976173423329066180576, −7.63556674028364416007324107725, −6.69943201088657051689996857993, −5.457909349559924798595425382885, −4.50071572305420217477658032656, −3.11770312281117310675445193738, −2.31594058556797201409257889043, −0.7776504084594345896269909462, 0.6425579942361580460664486614, 3.50619395679943048646673377021, 4.11359494513983684868592481019, 4.82622692589226975386170759792, 6.21704064247133190093553800108, 6.858943044568806088609944742588, 8.022928080494885177618271164086, 9.04852533892017153497382612820, 9.854106526138919849727771921392, 10.98627655090749168108543095639, 12.06362440997240352511989119529, 12.83763269373110014353952540144, 14.29576162465810560690369751678, 14.76187771117111019681570916922, 15.9099863203213505085063286273, 16.54882873729633890012225657629, 16.87430986204497826447691399279, 17.98503237398733204814208494716, 19.3034900207263217791186499829, 19.946948003922923717541454929926, 21.17178813352214676362079147097, 22.32254920200191910894810104595, 22.5690497860841469053072833136, 23.63333153321023719081532797214, 24.094355199531170303225331301330

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