Properties

Label 1-847-847.100-r0-0-0
Degree $1$
Conductor $847$
Sign $0.516 + 0.856i$
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  + (0.981 + 0.189i)2-s + (−0.5 + 0.866i)3-s + (0.928 + 0.371i)4-s + (−0.888 − 0.458i)5-s + (−0.654 + 0.755i)6-s + (0.841 + 0.540i)8-s + (−0.5 − 0.866i)9-s + (−0.786 − 0.618i)10-s + (−0.786 + 0.618i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.723 + 0.690i)16-s + (0.580 + 0.814i)17-s + (−0.327 − 0.945i)18-s + (0.580 − 0.814i)19-s + (−0.654 − 0.755i)20-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)2-s + (−0.5 + 0.866i)3-s + (0.928 + 0.371i)4-s + (−0.888 − 0.458i)5-s + (−0.654 + 0.755i)6-s + (0.841 + 0.540i)8-s + (−0.5 − 0.866i)9-s + (−0.786 − 0.618i)10-s + (−0.786 + 0.618i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.723 + 0.690i)16-s + (0.580 + 0.814i)17-s + (−0.327 − 0.945i)18-s + (0.580 − 0.814i)19-s + (−0.654 − 0.755i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.787286424 + 1.009559464i\)
\(L(\frac12)\) \(\approx\) \(1.787286424 + 1.009559464i\)
\(L(1)\) \(\approx\) \(1.423404642 + 0.4838115969i\)
\(L(1)\) \(\approx\) \(1.423404642 + 0.4838115969i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.981 + 0.189i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.888 - 0.458i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (0.580 + 0.814i)T \)
19 \( 1 + (0.580 - 0.814i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (0.928 - 0.371i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.327 + 0.945i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (0.981 - 0.189i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−22.307448983725995075296788340673, −21.31434907455235514632603718749, −20.36120896932754034120303371023, −19.59041338496849822223497850782, −18.776355616788209954776185812693, −18.42293907184566254603572665515, −16.887153309171958385348523171, −16.36103650063710240391968543699, −15.48046345709048812468654325127, −14.36519299880560722902497001099, −14.02403498188913823791012175889, −12.92865653666940313762224479740, −12.06555284124670034313180411499, −11.708365669590818272169488248109, −10.93730703423074254843718862089, −9.98275199056455558475826440985, −8.44082441482486939885861935815, −7.31530615351248697820577570549, −7.02366126846845911371398165500, −6.00061470419916843706680452016, −5.06917610407374261200746634712, −4.128572026330893554378782106199, −3.07335350931863783159373546884, −2.15811621443513919666945421101, −0.90667488125107485977089004429, 1.06378932130890352918696190319, 2.96355874818923892122377445266, 3.57650162650540694283767030365, 4.52072949388596831737319556864, 5.222082605526967094006579492604, 5.935105304276220750101074589466, 7.16429697736940511746375023534, 7.96412013667749426191369351438, 9.00481937411079533360039886951, 10.17038005827450295913456581298, 11.11718544289602624850581745733, 11.573912093129009083044895486622, 12.59830295277110220648243119877, 13.07112304017745976560288107483, 14.50787046603382893494720734198, 15.069626400849233758983648814, 15.71387978855186288682200286935, 16.40621038410651019477078784956, 17.07598335607251057012124486513, 17.99844695310945178697791948253, 19.55370206476427405796911954902, 20.005570597743677176432873796723, 20.84542835611368306824203046087, 21.53982725877902863689181314655, 22.34400987341438172933739183060

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