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Properties

Label 1-633-633.170-r1-0-0
Degree $1$
Conductor $633$
Sign $-0.404 - 0.914i$
Analytic cond. $68.0252$
Root an. cond. $68.0252$
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  + (0.873 + 0.486i)2-s + (0.525 + 0.850i)4-s + (0.393 + 0.919i)5-s + (0.0149 − 0.999i)7-s + (0.0448 + 0.998i)8-s + (−0.104 + 0.994i)10-s + (−0.753 − 0.657i)11-s + (−0.550 − 0.834i)13-s + (0.5 − 0.866i)14-s + (−0.447 + 0.894i)16-s + (0.772 + 0.635i)17-s + (−0.978 + 0.207i)19-s + (−0.575 + 0.817i)20-s + (−0.337 − 0.941i)22-s + (−0.309 − 0.951i)23-s + ⋯
L(s)  = 1  + (0.873 + 0.486i)2-s + (0.525 + 0.850i)4-s + (0.393 + 0.919i)5-s + (0.0149 − 0.999i)7-s + (0.0448 + 0.998i)8-s + (−0.104 + 0.994i)10-s + (−0.753 − 0.657i)11-s + (−0.550 − 0.834i)13-s + (0.5 − 0.866i)14-s + (−0.447 + 0.894i)16-s + (0.772 + 0.635i)17-s + (−0.978 + 0.207i)19-s + (−0.575 + 0.817i)20-s + (−0.337 − 0.941i)22-s + (−0.309 − 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $-0.404 - 0.914i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ -0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1731875865 - 0.2661209035i\)
\(L(\frac12)\) \(\approx\) \(0.1731875865 - 0.2661209035i\)
\(L(1)\) \(\approx\) \(1.289405065 + 0.4324975487i\)
\(L(1)\) \(\approx\) \(1.289405065 + 0.4324975487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.873 + 0.486i)T \)
5 \( 1 + (0.393 + 0.919i)T \)
7 \( 1 + (0.0149 - 0.999i)T \)
11 \( 1 + (-0.753 - 0.657i)T \)
13 \( 1 + (-0.550 - 0.834i)T \)
17 \( 1 + (0.772 + 0.635i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.998 + 0.0598i)T \)
31 \( 1 + (0.365 - 0.930i)T \)
37 \( 1 + (-0.946 - 0.323i)T \)
41 \( 1 + (-0.712 + 0.701i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (-0.971 - 0.237i)T \)
53 \( 1 + (-0.525 + 0.850i)T \)
59 \( 1 + (-0.251 - 0.967i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (-0.0448 + 0.998i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (0.995 + 0.0896i)T \)
97 \( 1 + (0.983 + 0.178i)T \)
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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

−22.96717992063858682618540684915, −21.995917826585906244273068386930, −21.205333633863721341207402716706, −20.930247767893952990843598111397, −19.84867384446631641720834302679, −19.06684646960260456576084368743, −18.219247974377375250109741727592, −17.1269224172885296528998722560, −16.1209690628283525577957876630, −15.41913302070369508029164660779, −14.568150143999250646354854352507, −13.62207095119693275592762950091, −12.81606519691192429897877239773, −12.14987832041011320487139650423, −11.5470058147482856730884620301, −10.16238128670864920738325592912, −9.55021635527443460937181385614, −8.60003153992751101099556054803, −7.29145278743164809863186215262, −6.1781502357658282914313816596, −5.11183222927931997620134594726, −4.89032182207142570976108599862, −3.474138173425985539963151504154, −2.224249205071295566929475754525, −1.67000824885436772093356255510, 0.046422492276705319316626767023, 1.98809350470000693431770115083, 3.07785162230739482214978871572, 3.78270424759355711302945201586, 5.00055279953591824149514945680, 5.95463762074003345073859904579, 6.69075652321571544947572167786, 7.69748864077050061788375699407, 8.24631864262877595241561223942, 10.06271961638314489034013447378, 10.59278764394849397488382283673, 11.444465780059953008018033568, 12.776762218469542829432298158403, 13.2506990241906970490543641921, 14.26370405200949572203345359148, 14.748537207870151197722474804560, 15.615363631802559360439240387202, 16.7965252810920399508448897897, 17.15508185749345105501877465066, 18.26694381173956484577536257294, 19.17602716718882638917099526617, 20.29087142484400198003144477713, 21.02630574985493129475867634677, 21.79457994919997293434872299144, 22.58790234638394576119032737573

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