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Properties

Label 1-421-421.49-r0-0-0
Degree $1$
Conductor $421$
Sign $0.929 - 0.369i$
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  + (−0.0448 − 0.998i)2-s + (0.753 + 0.657i)3-s + (−0.995 + 0.0896i)4-s + (−0.995 − 0.0896i)5-s + (0.623 − 0.781i)6-s + (0.753 − 0.657i)7-s + (0.134 + 0.990i)8-s + (0.134 + 0.990i)9-s + (−0.0448 + 0.998i)10-s + (−0.393 + 0.919i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.691 − 0.722i)14-s + (−0.691 − 0.722i)15-s + (0.983 − 0.178i)16-s + (−0.963 + 0.266i)17-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (0.753 + 0.657i)3-s + (−0.995 + 0.0896i)4-s + (−0.995 − 0.0896i)5-s + (0.623 − 0.781i)6-s + (0.753 − 0.657i)7-s + (0.134 + 0.990i)8-s + (0.134 + 0.990i)9-s + (−0.0448 + 0.998i)10-s + (−0.393 + 0.919i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.691 − 0.722i)14-s + (−0.691 − 0.722i)15-s + (0.983 − 0.178i)16-s + (−0.963 + 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.929 - 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.325007844 - 0.2536901932i\)
\(L(\frac12)\) \(\approx\) \(1.325007844 - 0.2536901932i\)
\(L(1)\) \(\approx\) \(1.091529367 - 0.2567685087i\)
\(L(1)\) \(\approx\) \(1.091529367 - 0.2567685087i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad421 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
3 \( 1 + (0.753 + 0.657i)T \)
5 \( 1 + (-0.995 - 0.0896i)T \)
7 \( 1 + (0.753 - 0.657i)T \)
11 \( 1 + (-0.393 + 0.919i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.963 + 0.266i)T \)
19 \( 1 + (0.936 + 0.351i)T \)
23 \( 1 + (0.858 - 0.512i)T \)
29 \( 1 + T \)
31 \( 1 + (0.473 + 0.880i)T \)
37 \( 1 + (-0.393 + 0.919i)T \)
41 \( 1 + (0.936 + 0.351i)T \)
43 \( 1 + (0.983 - 0.178i)T \)
47 \( 1 + (-0.995 - 0.0896i)T \)
53 \( 1 + (0.753 - 0.657i)T \)
59 \( 1 + (-0.550 + 0.834i)T \)
61 \( 1 + (0.983 - 0.178i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (-0.550 - 0.834i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.983 + 0.178i)T \)
83 \( 1 + (-0.550 + 0.834i)T \)
89 \( 1 + (0.473 - 0.880i)T \)
97 \( 1 + (0.936 - 0.351i)T \)
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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

−24.460096659954263986188339168979, −23.73791782758665772453693574022, −22.924265925524929139655269783669, −21.72195223420435257477631304662, −20.86481834354617992996875117301, −19.55163453467526140309084722130, −18.95248482495615888290912125671, −18.26519464068355823635655884348, −17.45173028609075000408077408915, −15.96409185461900895986836800611, −15.63834201431298042592997098562, −14.5928880617764400018646948230, −13.92624526247580481348015516442, −13.06306945144941008684491503982, −11.899726689584607685924692934487, −11.09722698121798061838587454699, −9.21919598552575957443674096244, −8.72281838333818820615658360293, −7.88481903047692820377869032322, −7.1707288691260979157755856075, −6.160208287184161907841505865440, −4.89311597170206574187621546252, −3.84855227823946293192497814116, −2.65351562417486823583926597950, −0.94385608000060134067310024837, 1.173454263410406050299132209624, 2.614850284239681146364370826443, 3.52750438377752663209236830227, 4.52776049965724399427380644279, 4.977032659541098309455826121070, 7.31889056280126424687689339928, 8.14726850724468517302713486790, 8.777671098906943537397417649929, 10.10944676991033321395043688216, 10.64542689243573454126857430919, 11.51654260942469528334796711511, 12.63327195007010030020303036777, 13.48062748320208346131294963728, 14.4972286307624024473474360024, 15.222397470280198784181526556189, 16.15091597117226896657594613459, 17.45373760706330116126813787952, 18.19506810989500600320844929845, 19.42102309243995145428773107213, 19.98949552893495974130842133052, 20.617410210623007565460352187244, 21.14407497721428039804543084711, 22.52476520180726498542661753602, 22.922626811847763960357145287881, 24.00655557622164054108107543974

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