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Properties

Label 1-421-421.247-r0-0-0
Degree $1$
Conductor $421$
Sign $-0.125 - 0.992i$
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  + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.222 + 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + 13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (0.623 − 0.781i)16-s + (−0.222 + 0.974i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.900 − 0.433i)5-s + (−0.222 + 0.974i)6-s + (−0.900 + 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)11-s + 12-s + 13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (0.623 − 0.781i)16-s + (−0.222 + 0.974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $-0.125 - 0.992i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ -0.125 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2768937352 - 0.3141524619i\)
\(L(\frac12)\) \(\approx\) \(0.2768937352 - 0.3141524619i\)
\(L(1)\) \(\approx\) \(0.4123641368 - 0.2561479650i\)
\(L(1)\) \(\approx\) \(0.4123641368 - 0.2561479650i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−24.04844976065093168244178685472, −23.568517071763339686315743256201, −22.77839170999032147679133676051, −22.48142895816548816434068288921, −21.18309839164976124100258709478, −19.94119673376321357170949195889, −19.00542498855544889266973351075, −18.0892794698227399426349139091, −17.528742423771466499773261882914, −16.12396839361539033613945271262, −15.92909178560585781034977554193, −15.3686430324788460770980738135, −13.97440964534416080823707814192, −13.0610865717184479336174605080, −11.994729042508533046439052865427, −10.79698320381490277198536731030, −10.19220287804625176357856853960, −9.14596913135466540460345159456, −7.92479713584069964463440466049, −6.90024888667827444352533866821, −6.413638548581648540635258761597, −5.112924615051236898393839433906, −4.2795771444819417592365861793, −3.22897380626284079779554252872, −0.65990317676712271619425340230, 0.545637798139749403240808007, 1.93325199601874345724571138270, 3.360817770804740033026968192438, 4.27358414215664499150622055267, 5.49617401205219742086681734334, 6.44649508260932296154301720102, 8.05102395627576824447709959266, 8.41999740671636461855571167562, 9.93657179317176651296272659798, 10.72405967703507201883419192872, 11.54550987619032742746404097718, 12.51439471770622321368729677640, 12.80108682807592471099047608471, 13.83712153116097840101026832589, 15.611931595037383505023121953484, 16.209802420177900521455586491422, 17.09100429148568714535309944833, 18.24079561744990688690052014712, 18.86443616473051069326240792610, 19.40349737464926750921020003398, 20.501596318753389551579333675328, 21.408778771047838611817387065942, 22.272775843505117222284916725716, 23.18930956393548828787633568196, 23.50825286094841330657581061188

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