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Properties

Label 1-899-899.314-r0-0-0
Degree $1$
Conductor $899$
Sign $0.973 + 0.227i$
Analytic cond. $4.17493$
Root an. cond. $4.17493$
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  + (−0.393 − 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 + 0.722i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.134 + 0.990i)7-s + (0.936 + 0.351i)8-s + (0.936 + 0.351i)9-s + (0.983 − 0.178i)10-s + (0.936 − 0.351i)11-s + (−0.809 + 0.587i)12-s + (−0.0448 − 0.998i)13-s + (0.858 − 0.512i)14-s + (−0.393 + 0.919i)15-s + (−0.0448 − 0.998i)16-s + (0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (0.983 + 0.178i)3-s + (−0.691 + 0.722i)4-s + (−0.222 + 0.974i)5-s + (−0.222 − 0.974i)6-s + (0.134 + 0.990i)7-s + (0.936 + 0.351i)8-s + (0.936 + 0.351i)9-s + (0.983 − 0.178i)10-s + (0.936 − 0.351i)11-s + (−0.809 + 0.587i)12-s + (−0.0448 − 0.998i)13-s + (0.858 − 0.512i)14-s + (−0.393 + 0.919i)15-s + (−0.0448 − 0.998i)16-s + (0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 899 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 899 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(899\)    =    \(29 \cdot 31\)
Sign: $0.973 + 0.227i$
Analytic conductor: \(4.17493\)
Root analytic conductor: \(4.17493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{899} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 899,\ (0:\ ),\ 0.973 + 0.227i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.793310560 + 0.2063609790i\)
\(L(\frac12)\) \(\approx\) \(1.793310560 + 0.2063609790i\)
\(L(1)\) \(\approx\) \(1.269667860 - 0.05999198274i\)
\(L(1)\) \(\approx\) \(1.269667860 - 0.05999198274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.393 - 0.919i)T \)
3 \( 1 + (0.983 + 0.178i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
7 \( 1 + (0.134 + 0.990i)T \)
11 \( 1 + (0.936 - 0.351i)T \)
13 \( 1 + (-0.0448 - 0.998i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.983 - 0.178i)T \)
23 \( 1 + (0.858 - 0.512i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.753 + 0.657i)T \)
47 \( 1 + (-0.0448 - 0.998i)T \)
53 \( 1 + (-0.995 + 0.0896i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.900 + 0.433i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.550 + 0.834i)T \)
73 \( 1 + (-0.995 - 0.0896i)T \)
79 \( 1 + (0.936 + 0.351i)T \)
83 \( 1 + (0.983 - 0.178i)T \)
89 \( 1 + (0.858 + 0.512i)T \)
97 \( 1 + (-0.691 + 0.722i)T \)
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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

−21.97988120988740790810700324256, −20.68706762240509588510511558333, −20.3328100222057783151198090898, −19.44292382765669098337045410558, −18.90587948540137987538820980349, −17.79981612469562205792691675746, −16.99153494767994171809321115351, −16.38796074232603512945870612593, −15.6308671670979007448384786050, −14.66776236432995655735343664122, −13.867625825501371177978443275239, −13.5809048772295039540269596274, −12.42093552645497326871609336960, −11.41453196210243774223442484464, −9.97702486949272124650380285963, −9.31379568874975934275737284060, −8.88623993892932679323322702066, −7.66385985358619354027887226894, −7.367995980195485066921621520332, −6.416569209386764593951022500279, −4.94495468575924527962191035106, −4.365454873939301611759897777906, −3.45521641012917138634994221132, −1.5930031774258921851144511659, −1.0156695790958479215795375432, 1.300396681471082148267640237957, 2.40212595016376278258648150262, 3.13700333042940067948707528189, 3.67659040904498128905058453787, 4.90795339567626597944456154565, 6.22658005244608325988031539989, 7.49124702492160678653866756592, 8.15468458073648093933341412552, 8.97870778526038027496055581688, 9.68639910246024867887402457178, 10.5657120804985995848444284109, 11.30431001835996806262026644517, 12.23837489184589299522694873262, 13.0003811977753084066094318680, 14.01493765722322523981110534952, 14.72036408787656142454050805550, 15.32696035053907310402433436997, 16.40405490683030306039446599849, 17.584829661053473766769997298024, 18.33851454938859938811129674529, 18.95369167197873942386297601638, 19.56552828863538581144972992328, 20.218463373638008296971419717454, 21.19242108095293894403577291823, 21.90612661196631372616579462667

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