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Properties

Label 1-633-633.260-r1-0-0
Degree $1$
Conductor $633$
Sign $0.492 - 0.870i$
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  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (0.0448 + 0.998i)5-s + (0.998 + 0.0598i)7-s + (−0.983 + 0.178i)8-s + (0.913 + 0.406i)10-s + (0.963 − 0.266i)11-s + (−0.691 − 0.722i)13-s + (0.5 − 0.866i)14-s + (−0.280 + 0.959i)16-s + (0.925 − 0.379i)17-s + (0.669 − 0.743i)19-s + (0.772 − 0.635i)20-s + (0.193 − 0.981i)22-s + (−0.309 + 0.951i)23-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)2-s + (−0.599 − 0.800i)4-s + (0.0448 + 0.998i)5-s + (0.998 + 0.0598i)7-s + (−0.983 + 0.178i)8-s + (0.913 + 0.406i)10-s + (0.963 − 0.266i)11-s + (−0.691 − 0.722i)13-s + (0.5 − 0.866i)14-s + (−0.280 + 0.959i)16-s + (0.925 − 0.379i)17-s + (0.669 − 0.743i)19-s + (0.772 − 0.635i)20-s + (0.193 − 0.981i)22-s + (−0.309 + 0.951i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 633 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(633\)    =    \(3 \cdot 211\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(68.0252\)
Root analytic conductor: \(68.0252\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{633} (260, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 633,\ (1:\ ),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.574968473 - 1.500818600i\)
\(L(\frac12)\) \(\approx\) \(2.574968473 - 1.500818600i\)
\(L(1)\) \(\approx\) \(1.400769173 - 0.5809903172i\)
\(L(1)\) \(\approx\) \(1.400769173 - 0.5809903172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.447 - 0.894i)T \)
5 \( 1 + (0.0448 + 0.998i)T \)
7 \( 1 + (0.998 + 0.0598i)T \)
11 \( 1 + (0.963 - 0.266i)T \)
13 \( 1 + (-0.691 - 0.722i)T \)
17 \( 1 + (0.925 - 0.379i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.971 + 0.237i)T \)
31 \( 1 + (0.0747 + 0.997i)T \)
37 \( 1 + (0.251 + 0.967i)T \)
41 \( 1 + (0.999 + 0.0299i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (-0.575 - 0.817i)T \)
53 \( 1 + (0.599 - 0.800i)T \)
59 \( 1 + (-0.525 + 0.850i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.955 + 0.294i)T \)
79 \( 1 + (0.983 + 0.178i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.936 - 0.351i)T \)
97 \( 1 + (0.753 + 0.657i)T \)
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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

−22.97383449156376811597168066001, −22.18868625521124639979564788575, −21.16007363342268338915902255570, −20.776190250558551899854356206930, −19.669069505126041869657368109028, −18.52781097440493599774676268020, −17.60608851789939989915160719537, −16.72559084862008314276576855638, −16.59589613718012147064695719174, −15.2791448915568231765935868388, −14.33158559968776569801583868188, −14.10921220428555639039975476116, −12.69118724427718161801954213708, −12.20366105394834889281889724121, −11.356302782245920611995597676992, −9.66907994007551788858284086412, −9.102515715357984489133373575889, −7.97943877670262161477184006685, −7.53585958430335723316632177360, −6.17878376887013274682431525970, −5.3966943290370453068665300642, −4.41893060105629248177057297650, −3.91784863345444738965495116921, −2.10697554294377918736877702266, −0.884034448643990490573404103855, 0.84128391609853911503848810280, 1.93296547037648689480238185924, 3.00697335810207503839281343005, 3.75363664867416075256352390415, 5.02628254637509310020519261850, 5.712246085830158241931375573, 6.98580407880331763138455947154, 7.92806006389693336797466963651, 9.22222816233982446434934644737, 9.98013891203691738343967727178, 10.931753334244445827903299111543, 11.573571447723372115126949674867, 12.20314371572050341925161733607, 13.507497061270619857490201551415, 14.249098009388461104875571420835, 14.73778306533737314464967127084, 15.568632648106955855094848088254, 17.13491452464396649525074223267, 17.89028398396772789219637775688, 18.51247315845941812947130790702, 19.50062242814601607882225429401, 20.057727599522204140870637788064, 21.119735503626120871295915011399, 21.81132468851076608207282247040, 22.40435985744060668144725265503

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