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Properties

Label 1-669-669.245-r0-0-0
Degree $1$
Conductor $669$
Sign $0.0141 + 0.999i$
Analytic cond. $3.10682$
Root an. cond. $3.10682$
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  + (0.292 + 0.956i)2-s + (−0.828 + 0.559i)4-s + (−0.933 − 0.359i)5-s + (−0.996 + 0.0848i)7-s + (−0.778 − 0.628i)8-s + (0.0706 − 0.997i)10-s + (0.475 − 0.879i)11-s + (0.967 + 0.251i)13-s + (−0.372 − 0.927i)14-s + (0.372 − 0.927i)16-s + (−0.524 − 0.851i)17-s + (−0.639 + 0.769i)19-s + (0.974 − 0.224i)20-s + (0.980 + 0.196i)22-s + (0.155 + 0.987i)23-s + ⋯
L(s)  = 1  + (0.292 + 0.956i)2-s + (−0.828 + 0.559i)4-s + (−0.933 − 0.359i)5-s + (−0.996 + 0.0848i)7-s + (−0.778 − 0.628i)8-s + (0.0706 − 0.997i)10-s + (0.475 − 0.879i)11-s + (0.967 + 0.251i)13-s + (−0.372 − 0.927i)14-s + (0.372 − 0.927i)16-s + (−0.524 − 0.851i)17-s + (−0.639 + 0.769i)19-s + (0.974 − 0.224i)20-s + (0.980 + 0.196i)22-s + (0.155 + 0.987i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0141 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(669\)    =    \(3 \cdot 223\)
Sign: $0.0141 + 0.999i$
Analytic conductor: \(3.10682\)
Root analytic conductor: \(3.10682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{669} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 669,\ (0:\ ),\ 0.0141 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6851010131 + 0.6948813025i\)
\(L(\frac12)\) \(\approx\) \(0.6851010131 + 0.6948813025i\)
\(L(1)\) \(\approx\) \(0.7771135076 + 0.4060337862i\)
\(L(1)\) \(\approx\) \(0.7771135076 + 0.4060337862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 \)
good2 \( 1 + (0.292 + 0.956i)T \)
5 \( 1 + (-0.933 - 0.359i)T \)
7 \( 1 + (-0.996 + 0.0848i)T \)
11 \( 1 + (0.475 - 0.879i)T \)
13 \( 1 + (0.967 + 0.251i)T \)
17 \( 1 + (-0.524 - 0.851i)T \)
19 \( 1 + (-0.639 + 0.769i)T \)
23 \( 1 + (0.155 + 0.987i)T \)
29 \( 1 + (0.990 - 0.141i)T \)
31 \( 1 + (0.998 + 0.0565i)T \)
37 \( 1 + (-0.999 - 0.0282i)T \)
41 \( 1 + (0.127 + 0.991i)T \)
43 \( 1 + (-0.951 - 0.306i)T \)
47 \( 1 + (0.639 + 0.769i)T \)
53 \( 1 + (0.886 - 0.462i)T \)
59 \( 1 + (0.660 + 0.750i)T \)
61 \( 1 + (-0.702 + 0.712i)T \)
67 \( 1 + (0.0706 + 0.997i)T \)
71 \( 1 + (-0.980 + 0.196i)T \)
73 \( 1 + (-0.759 - 0.649i)T \)
79 \( 1 + (0.795 - 0.605i)T \)
83 \( 1 + (0.951 - 0.306i)T \)
89 \( 1 + (0.933 - 0.359i)T \)
97 \( 1 + (-0.155 + 0.987i)T \)
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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

−22.57456638244492900658393894163, −21.907878150126148422643297070350, −20.81080238477084959958017441270, −19.965230704295979111326698944961, −19.48802705052278771786781008128, −18.795760918052383131081005847, −17.884794118813029704183248819590, −16.93138815395736808795491248430, −15.57913979708396253414668072380, −15.24283978706522089043680734000, −14.11758839468381027995754513508, −13.16289934053168195371221623002, −12.45898803796654370334160765842, −11.78253806109687236151209887042, −10.64417488923907887089000200188, −10.307964286346708080278190993730, −8.98197089961211352576580824669, −8.386923845637125069116628493502, −6.84678497806701746211475893697, −6.27236354407714370174753047002, −4.71633415267495239465666266749, −3.97782826376347471849389106782, −3.20630668678626958607760094936, −2.15544526341494217320108867748, −0.63450631009322735646434280600, 0.86755952530120222809098303922, 3.104609133507136806122870913773, 3.75313750714829799838520979001, 4.65513378312395512439931971272, 5.89388005059217966331175222671, 6.54881801001495120851549571483, 7.47312821495159063409319001718, 8.61251362640917292253925964539, 8.9047600880229704359377636070, 10.17417621650585127994161390887, 11.55748670406116681106245054942, 12.14826066585761791840997672198, 13.30196028028930734684766175601, 13.69870395687034419060878587519, 14.90094007453274003781055049758, 15.81814199328169323217956504390, 16.15043403137066683360314015476, 16.87249750566495509081245628929, 17.974540975905988846756747128697, 19.03892873210620175957643478975, 19.3754275893202082510676615178, 20.6621625430266745263804187614, 21.533410209267423161825692677265, 22.47452297773581304691923969044, 23.15516137861209796683032010804

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