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Properties

Label 1-421-421.139-r0-0-0
Degree $1$
Conductor $421$
Sign $0.975 + 0.221i$
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  + (−0.963 − 0.266i)2-s + (−0.393 + 0.919i)3-s + (0.858 + 0.512i)4-s + (0.858 − 0.512i)5-s + (0.623 − 0.781i)6-s + (−0.393 − 0.919i)7-s + (−0.691 − 0.722i)8-s + (−0.691 − 0.722i)9-s + (−0.963 + 0.266i)10-s + (0.753 + 0.657i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.134 + 0.990i)14-s + (0.134 + 0.990i)15-s + (0.473 + 0.880i)16-s + (−0.0448 + 0.998i)17-s + ⋯
L(s)  = 1  + (−0.963 − 0.266i)2-s + (−0.393 + 0.919i)3-s + (0.858 + 0.512i)4-s + (0.858 − 0.512i)5-s + (0.623 − 0.781i)6-s + (−0.393 − 0.919i)7-s + (−0.691 − 0.722i)8-s + (−0.691 − 0.722i)9-s + (−0.963 + 0.266i)10-s + (0.753 + 0.657i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.134 + 0.990i)14-s + (0.134 + 0.990i)15-s + (0.473 + 0.880i)16-s + (−0.0448 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(421\)
Sign: $0.975 + 0.221i$
Analytic conductor: \(1.95511\)
Root analytic conductor: \(1.95511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{421} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 421,\ (0:\ ),\ 0.975 + 0.221i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8406837554 + 0.09444637477i\)
\(L(\frac12)\) \(\approx\) \(0.8406837554 + 0.09444637477i\)
\(L(1)\) \(\approx\) \(0.7336113885 + 0.04885904538i\)
\(L(1)\) \(\approx\) \(0.7336113885 + 0.04885904538i\)

Euler product

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

−24.67785800446082235029669655381, −23.43270758118786986692082700332, −22.50975689759027448091406669452, −21.713476225493596643973426916553, −20.509814327214462729247098605363, −19.432275208208663670325999816016, −18.76485145041921430479814962458, −18.10560178837012319386816931783, −17.51068339256113258586286462983, −16.55337624523464845887206754453, −15.66991197068352595998978316952, −14.46149411636223416524941266885, −13.72629559398157533723120021740, −12.44916317742174268631808025610, −11.6725126731679090902966313490, −10.68727816454145489783401337233, −9.77270309594752763052371844504, −8.7414328157347251669122735975, −7.92403502052376342571650422412, −6.67561754400168090371507764723, −6.09526644003245630273810831066, −5.49270112648982370707472587832, −2.97233732462581335347470019454, −2.19417450988180789499338451070, −0.95538801788433048454481006788, 0.984894022586756821428779233717, 2.244525624620616175478533323267, 3.81564780229676061751202204302, 4.55431179793200491886477729782, 6.271472356059902562356861453417, 6.65407269052401613623798001988, 8.336374886214056161306410539799, 9.19182215302608832681177973973, 9.93282603333721220355338680334, 10.47790962381856065397395358922, 11.560413960454669171332176371786, 12.463881088384265408353115613358, 13.63746532261329018635439449008, 14.75677927498237259882716055059, 15.92485351641371351806284837610, 16.609420742089756213630370091800, 17.35836460534144285896021478480, 17.66601032958252471035257158756, 19.222154618006150809157162061701, 20.026831214003994834614489955706, 20.69323792344084406742122522639, 21.55796422840560575662845193279, 22.12783300987587793038583553250, 23.444358764815085117385371087238, 24.29090578125481757391006336903

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