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Properties

Label 1-421-421.321-r0-0-0
Degree $1$
Conductor $421$
Sign $0.987 + 0.160i$
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  + (0.473 − 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 − 0.834i)4-s + (−0.550 + 0.834i)5-s + (−0.900 − 0.433i)6-s + (−0.0448 + 0.998i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.473 + 0.880i)10-s + (−0.963 − 0.266i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.858 + 0.512i)14-s + (0.858 + 0.512i)15-s + (−0.393 + 0.919i)16-s + (0.983 − 0.178i)17-s + ⋯
L(s)  = 1  + (0.473 − 0.880i)2-s + (−0.0448 − 0.998i)3-s + (−0.550 − 0.834i)4-s + (−0.550 + 0.834i)5-s + (−0.900 − 0.433i)6-s + (−0.0448 + 0.998i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 + 0.0896i)9-s + (0.473 + 0.880i)10-s + (−0.963 − 0.266i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.858 + 0.512i)14-s + (0.858 + 0.512i)15-s + (−0.393 + 0.919i)16-s + (0.983 − 0.178i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8616127088 + 0.06969047804i\)
\(L(\frac12)\) \(\approx\) \(0.8616127088 + 0.06969047804i\)
\(L(1)\) \(\approx\) \(0.8584131971 - 0.3538997031i\)
\(L(1)\) \(\approx\) \(0.8584131971 - 0.3538997031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (0.473 - 0.880i)T \)
3 \( 1 + (-0.0448 - 0.998i)T \)
5 \( 1 + (-0.550 + 0.834i)T \)
7 \( 1 + (-0.0448 + 0.998i)T \)
11 \( 1 + (-0.963 - 0.266i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.983 - 0.178i)T \)
19 \( 1 + (-0.691 + 0.722i)T \)
23 \( 1 + (0.936 - 0.351i)T \)
29 \( 1 + T \)
31 \( 1 + (0.753 + 0.657i)T \)
37 \( 1 + (-0.963 - 0.266i)T \)
41 \( 1 + (-0.691 + 0.722i)T \)
43 \( 1 + (-0.393 + 0.919i)T \)
47 \( 1 + (-0.550 + 0.834i)T \)
53 \( 1 + (-0.0448 + 0.998i)T \)
59 \( 1 + (0.134 + 0.990i)T \)
61 \( 1 + (-0.393 + 0.919i)T \)
67 \( 1 + (0.309 - 0.951i)T \)
71 \( 1 + (0.134 - 0.990i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.393 - 0.919i)T \)
83 \( 1 + (0.134 + 0.990i)T \)
89 \( 1 + (0.753 - 0.657i)T \)
97 \( 1 + (-0.691 - 0.722i)T \)
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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

−23.907663677015994639359598595908, −23.245633422999035912666547561804, −22.88832716101495444777799318921, −21.56886656281112143696213817606, −20.80205874145797795746671359629, −20.31659904667079256218166169111, −19.06206844981141405611512733082, −17.39705673284669777524356439474, −17.1693018342513467013979590119, −16.09293299682846624194203685016, −15.56888052885718890960786705923, −14.84107299367093485819089362723, −13.607437806165648281769627289731, −12.960732556128966977851828144963, −11.883193349552450530075602628274, −10.66336250743422128390098251151, −9.80430489364802113006198555164, −8.5031289409394832461753435155, −8.02059540434562727879962459161, −6.85488474990012580898760653876, −5.36491305456406159353227160206, −4.92422139287130293318062715300, −3.89600788924426282531718774845, −3.10833810916352015027367387392, −0.442143948373012182196206604960, 1.45966266161416679595504792828, 2.66883036893320725734238718778, 3.1740212937465702531365409453, 4.77108666142168551468209810922, 5.91987438472577557074446935967, 6.65495301724896673542476895631, 8.01979810081149023570914420230, 8.81854167467842488164432927082, 10.23227816806120581394647653424, 11.110356793274849437959333581883, 11.97340109741379875018142371829, 12.45214014777763701512654452001, 13.56424464603013206888598212805, 14.374200048287885429124831576567, 15.11121711720088900610566569197, 16.26955301409576039518489148628, 17.82116638461716147834573678244, 18.66386612652650372270593489997, 18.89063791790732311978157374125, 19.60418119733351828935996221132, 21.035478514132789725669497018055, 21.4754158331137264535984788029, 22.79501229156152715613450237842, 23.16934097663866576782714085365, 23.918084401153735387683658383966

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