Properties

Label 1-684-684.283-r1-0-0
Degree $1$
Conductor $684$
Sign $0.755 - 0.654i$
Analytic cond. $73.5060$
Root an. cond. $73.5060$
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  + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)7-s − 11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s − 31-s + (−0.173 − 0.984i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.173 − 0.984i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)5-s + (0.5 + 0.866i)7-s − 11-s + (−0.939 + 0.342i)13-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)23-s + (0.766 + 0.642i)25-s + (0.173 − 0.984i)29-s − 31-s + (−0.173 − 0.984i)35-s + 37-s + (0.766 − 0.642i)41-s + (−0.173 − 0.984i)43-s + (−0.173 + 0.984i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(73.5060\)
Root analytic conductor: \(73.5060\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 684,\ (1:\ ),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8195936879 - 0.3056641634i\)
\(L(\frac12)\) \(\approx\) \(0.8195936879 - 0.3056641634i\)
\(L(1)\) \(\approx\) \(0.7629066717 + 0.02295059559i\)
\(L(1)\) \(\approx\) \(0.7629066717 + 0.02295059559i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.939 - 0.342i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.766 - 0.642i)T \)
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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

−22.67838774954493121523018611430, −21.87980211077776979287810841752, −20.87509006356302169171375809178, −19.890254614259228526279658633882, −19.71108274624281871195932978454, −18.30853789248183485729330507973, −17.9381309045398468521870179089, −16.689034475235860629281872476033, −16.12578554048794808974796041141, −14.90931050288874855389154031044, −14.67507370277737554800266694821, −13.36141818711538676646671341105, −12.65438433308569885114914109832, −11.62278383912865985308686662827, −10.72634381789559406112655362691, −10.323002843031343693420038455018, −8.89449789229145274210451973762, −7.85940796186507324596903509076, −7.43768321435120721455497061360, −6.41997936123625080399973077329, −4.959689229110844933066215031771, −4.3740012568914825362656478842, −3.240399765673451593521534616200, −2.20946702711478757972665189695, −0.62322893380308271629810156062, 0.3287051632575334646403450751, 1.97113855502436959055655488750, 2.85097777218757960831439897902, 4.210635720441874210471919225934, 4.96006691803348446525530778018, 5.8341775251052280203494463952, 7.28757791619958953746299398450, 7.83237798860463986329999668413, 8.808446183270720758993883259684, 9.56717558125881560462118878580, 10.88386290973131084861897133733, 11.57319111186189792164158138938, 12.314125981737682627808445615723, 13.11702988696550973971886438387, 14.22293370747002173933256164960, 15.32668944944952261831775068164, 15.54816641442877043899324987810, 16.573917661034820437762633809046, 17.59184126549980764872589252948, 18.377806059516930339276202119134, 19.185725991496688296426698030422, 19.93185797475400630829388848786, 20.79166985882150356089065924352, 21.60544182915979967388895327697, 22.36055413751386495360205786348

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