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Properties

Label 1-451-451.344-r0-0-0
Degree $1$
Conductor $451$
Sign $0.874 + 0.484i$
Analytic cond. $2.09443$
Root an. cond. $2.09443$
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  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.809 + 0.587i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + 17-s + (0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + 7-s + (0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.309 − 0.951i)10-s + (0.309 + 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.809 + 0.587i)14-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + 17-s + (0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(451\)    =    \(11 \cdot 41\)
Sign: $0.874 + 0.484i$
Analytic conductor: \(2.09443\)
Root analytic conductor: \(2.09443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{451} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 451,\ (0:\ ),\ 0.874 + 0.484i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5474244701 + 0.1415192195i\)
\(L(\frac12)\) \(\approx\) \(0.5474244701 + 0.1415192195i\)
\(L(1)\) \(\approx\) \(0.5198198820 + 0.1926247692i\)
\(L(1)\) \(\approx\) \(0.5198198820 + 0.1926247692i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + T \)
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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

−23.950809755302417097740075401, −23.17306617401904395322994442974, −22.092164765114952272839797801099, −21.26976928157031782771962469578, −20.22985195477956186211748433345, −19.67061839461294522884416872510, −18.57071238117133857741290817497, −18.07591045235528000901716290530, −17.02786431218248754443572814229, −16.58845373255625342758675650464, −15.55689089410363590784153898940, −14.25066208279281540860411954493, −12.8885761112553255473480783713, −12.12342274887910020498104499962, −11.74783499015463669851810856096, −10.74290471421902802403817629654, −9.88974108094085499710267012611, −8.44677379242783369499816579856, −7.79901257837130366966237247909, −7.22593171709483923101789212030, −5.58198957716495068714177316417, −4.66714943518312307400896285704, −3.38763181943296285494206366949, −1.81029313805121523193738684105, −0.972362903457358833852689793307, 0.62402831627941885280356704224, 2.29861047063563566003021003628, 4.05452984227669382650531896085, 4.94602965786706373292269383263, 5.93293202643182522498567410324, 7.08954057216488931615836700844, 7.70362620648752975161626956626, 8.8600646524053686774737574833, 9.92876157978712963139284687638, 10.70253356308817396886731018873, 11.54638607164032592572991869966, 12.05866615596255203292292123761, 14.092490181767433682182515067778, 14.828143255386008770990404090309, 15.44350890688926605597549996099, 16.40690140830056339847051280497, 17.06921435473625647937970518336, 18.00352106376397619803102964830, 18.58817313956104355898231825212, 19.6284509507085015052256501584, 20.53281515492054454966487664641, 21.594578748235320222606984266040, 22.478165624986071946663759016386, 23.51015623328084731720411817392, 23.891133350111606584850546739385

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