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Properties

Label 1-847-847.549-r0-0-0
Degree $1$
Conductor $847$
Sign $0.992 - 0.119i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (0.888 − 0.458i)2-s + (0.5 + 0.866i)3-s + (0.580 − 0.814i)4-s + (−0.928 − 0.371i)5-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.723 + 0.690i)17-s + (−0.0475 + 0.998i)18-s + (0.723 − 0.690i)19-s + (−0.841 + 0.540i)20-s + ⋯
L(s)  = 1  + (0.888 − 0.458i)2-s + (0.5 + 0.866i)3-s + (0.580 − 0.814i)4-s + (−0.928 − 0.371i)5-s + (0.841 + 0.540i)6-s + (0.142 − 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.723 + 0.690i)17-s + (−0.0475 + 0.998i)18-s + (0.723 − 0.690i)19-s + (−0.841 + 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.992 - 0.119i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.992 - 0.119i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.665263229 - 0.1599789956i\)
\(L(\frac12)\) \(\approx\) \(2.665263229 - 0.1599789956i\)
\(L(1)\) \(\approx\) \(1.852802998 - 0.1272228302i\)
\(L(1)\) \(\approx\) \(1.852802998 - 0.1272228302i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.888 - 0.458i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.928 - 0.371i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (0.723 - 0.690i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.995 - 0.0950i)T \)
37 \( 1 + (0.580 + 0.814i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.0475 - 0.998i)T \)
53 \( 1 + (-0.327 + 0.945i)T \)
59 \( 1 + (0.888 + 0.458i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.142 - 0.989i)T \)
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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

−22.61264887401763247793246233672, −21.25350732971360070787355069561, −20.538118689149731240242521896719, −19.81003546580251643742716178286, −19.07112928504646184077743150690, −18.066686436360036099021888344295, −17.455449461998304202038818709810, −16.00743352454867255760507477238, −15.766642556752880076687038712203, −14.587854557966994271075536110380, −14.208045200174310958421966562167, −13.27024773263965815577017714982, −12.44361984182414708381456535640, −11.83516293473997412219352707275, −11.096040878623075981677577294158, −9.720513606211209039687244551617, −8.27695096841808525347469349280, −7.87117223918715088457389014838, −7.198793383034147923520504055013, −6.234547772839383325486317363765, −5.3997456757108989123859509107, −4.082111480670581224576003416517, −3.24379797984379143815477852169, −2.65565257077240591768083470544, −1.08256538252437976866641300153, 1.14222368117143563488864406374, 2.560389151833762448872601741616, 3.39918382854204222676225134131, 4.28037725770024826801970236991, 4.720497674333752570331320387486, 5.85222058202990422395843628696, 6.989423143764568164923293417195, 8.11962288933452261082312551124, 8.94028404842132193492862117354, 9.95190774319864698593022855655, 10.72916573620227151210214523906, 11.60769660074748837936151781624, 12.18335120481265961057918845645, 13.303662625781861561729015202870, 14.06542594828835086162031346481, 14.81682782296364273322034771753, 15.56204977748857894998839133308, 16.18428935519545255162742361641, 16.84762628319314869938989553303, 18.50252379847054806093520576844, 19.32247390895647507559985124694, 19.87049096261909761239711011589, 20.62325773180547433223080485263, 21.24617978171911076482820833467, 21.98463135545705556855253520932

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