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Properties

Label 1-847-847.321-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.993 + 0.111i$
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  + (−0.696 + 0.717i)2-s + (−0.309 + 0.951i)3-s + (−0.0285 − 0.999i)4-s + (0.870 + 0.491i)5-s + (−0.466 − 0.884i)6-s + (0.736 + 0.676i)8-s + (−0.809 − 0.587i)9-s + (−0.959 + 0.281i)10-s + (0.959 + 0.281i)12-s + (0.610 + 0.791i)13-s + (−0.736 + 0.676i)15-s + (−0.998 + 0.0570i)16-s + (0.0855 − 0.996i)17-s + (0.985 − 0.170i)18-s + (−0.921 + 0.389i)19-s + (0.466 − 0.884i)20-s + ⋯
L(s)  = 1  + (−0.696 + 0.717i)2-s + (−0.309 + 0.951i)3-s + (−0.0285 − 0.999i)4-s + (0.870 + 0.491i)5-s + (−0.466 − 0.884i)6-s + (0.736 + 0.676i)8-s + (−0.809 − 0.587i)9-s + (−0.959 + 0.281i)10-s + (0.959 + 0.281i)12-s + (0.610 + 0.791i)13-s + (−0.736 + 0.676i)15-s + (−0.998 + 0.0570i)16-s + (0.0855 − 0.996i)17-s + (0.985 − 0.170i)18-s + (−0.921 + 0.389i)19-s + (0.466 − 0.884i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.993 + 0.111i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.993 + 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04823046124 + 0.8631117494i\)
\(L(\frac12)\) \(\approx\) \(0.04823046124 + 0.8631117494i\)
\(L(1)\) \(\approx\) \(0.5396534318 + 0.5306014687i\)
\(L(1)\) \(\approx\) \(0.5396534318 + 0.5306014687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.696 + 0.717i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.870 + 0.491i)T \)
13 \( 1 + (0.610 + 0.791i)T \)
17 \( 1 + (0.0855 - 0.996i)T \)
19 \( 1 + (-0.921 + 0.389i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (-0.516 + 0.856i)T \)
31 \( 1 + (-0.941 - 0.336i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (0.897 + 0.441i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.985 + 0.170i)T \)
53 \( 1 + (-0.998 - 0.0570i)T \)
59 \( 1 + (-0.897 + 0.441i)T \)
61 \( 1 + (0.696 + 0.717i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.974 + 0.226i)T \)
73 \( 1 + (-0.254 + 0.967i)T \)
79 \( 1 + (-0.198 + 0.980i)T \)
83 \( 1 + (-0.362 + 0.931i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.870 - 0.491i)T \)
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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

−21.63094119155870035578375536636, −20.71954188069121557183264189993, −20.1485811907930435038944801668, −19.13885368328045044041701100321, −18.62013912992913304170342028198, −17.69562365501256480169996141955, −17.1975486382498669992672812574, −16.68257722294068319357123335186, −15.413364570636739509367897919036, −14.104893525998183603658389376201, −13.10804050341492249894189502720, −12.87886201582882937322962997403, −12.061343016343730208865196041595, −10.82027008534088700075760475037, −10.5506310137041778205763586309, −9.207920649079401443840487166280, −8.5564243640177851271007056089, −7.80582653669053830950350771314, −6.67624026109168240385111664861, −5.90860726369047847499884846239, −4.803763166576290950786831499664, −3.42575209899736715280639569600, −2.27149858646964682865013862827, −1.60506715014504865531895198829, −0.536743787906569026841000602274, 1.35167962114571863768394297412, 2.58871426054036835562629974204, 3.91738335427904982513060983582, 5.0563828835921508602916527676, 5.76273218364168169128183023782, 6.54743873263365249687971548867, 7.39406344106895515116068098926, 8.8330225224646410588756128034, 9.245590917851691769238363065526, 10.0149684153158739309461103827, 10.94406276220030002439495172717, 11.30931934702561709837357912559, 12.93723835822320520500889036239, 14.10994758341052795565383675699, 14.49674863518171381151809409329, 15.38977118664557405857829537468, 16.263266868387932440383721146545, 16.80735064354785585962360339505, 17.58074026220683443738823897526, 18.33217720032933315491312180569, 19.02269490743749313506669415290, 20.16097744431389480959040167550, 21.01035516037932986722308405816, 21.61990743099040348550709664364, 22.66111979311360873554501568767

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