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Properties

Label 1-847-847.191-r0-0-0
Degree $1$
Conductor $847$
Sign $0.328 - 0.944i$
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  + (−0.851 + 0.524i)2-s + (0.913 − 0.406i)3-s + (0.449 − 0.893i)4-s + (0.483 − 0.875i)5-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (0.669 − 0.743i)9-s + (0.0475 + 0.998i)10-s + (0.0475 − 0.998i)12-s + (−0.254 + 0.967i)13-s + (0.0855 − 0.996i)15-s + (−0.595 − 0.803i)16-s + (0.640 − 0.768i)17-s + (−0.179 + 0.983i)18-s + (−0.969 − 0.244i)19-s + (−0.564 − 0.825i)20-s + ⋯
L(s)  = 1  + (−0.851 + 0.524i)2-s + (0.913 − 0.406i)3-s + (0.449 − 0.893i)4-s + (0.483 − 0.875i)5-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (0.669 − 0.743i)9-s + (0.0475 + 0.998i)10-s + (0.0475 − 0.998i)12-s + (−0.254 + 0.967i)13-s + (0.0855 − 0.996i)15-s + (−0.595 − 0.803i)16-s + (0.640 − 0.768i)17-s + (−0.179 + 0.983i)18-s + (−0.969 − 0.244i)19-s + (−0.564 − 0.825i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.146432296 - 0.8147062550i\)
\(L(\frac12)\) \(\approx\) \(1.146432296 - 0.8147062550i\)
\(L(1)\) \(\approx\) \(1.034388148 - 0.2313611660i\)
\(L(1)\) \(\approx\) \(1.034388148 - 0.2313611660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.851 + 0.524i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (0.483 - 0.875i)T \)
13 \( 1 + (-0.254 + 0.967i)T \)
17 \( 1 + (0.640 - 0.768i)T \)
19 \( 1 + (-0.969 - 0.244i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (-0.466 - 0.884i)T \)
31 \( 1 + (-0.935 + 0.353i)T \)
37 \( 1 + (0.988 - 0.151i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.179 - 0.983i)T \)
53 \( 1 + (-0.595 + 0.803i)T \)
59 \( 1 + (0.380 + 0.924i)T \)
61 \( 1 + (0.879 - 0.475i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.861 - 0.508i)T \)
79 \( 1 + (0.123 + 0.992i)T \)
83 \( 1 + (-0.736 - 0.676i)T \)
89 \( 1 + (-0.786 + 0.618i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−21.878229715298383859875163239800, −21.374461211962857708557494529889, −20.6234966107134950993356518254, −19.767806341713894377461310540992, −19.12726823302753649063728627670, −18.49082929494988956749225718194, −17.58745443189778268365487064270, −16.8541691544441763186154705564, −15.83565733395032801576432279038, −14.917248525040608912402384406110, −14.46480362296007217036480750863, −13.080528933534611460787119478511, −12.767183601712785514548148583932, −11.19495639479496203665000382373, −10.6552937906794124819173715971, −9.88149203115666877584770238633, −9.29760216517226931340407490902, −8.19888758856529170507903656177, −7.6300617295948618048922809428, −6.67121682892140839395235084668, −5.43603678883238852798637213804, −3.88475675111187864074511742368, −3.234012259143567857061608195509, −2.38712120849452165773719946220, −1.50013315774420348029533978616, 0.75051995551404658526664952707, 1.84088650071945608332616260458, 2.54284837032369662457122532370, 4.172958950204818492915511571367, 5.16251453698499483225381565078, 6.3083646261598136705789614028, 7.05636817619625936914479892970, 7.99630642948219589097833446202, 8.7442857592042625111226217792, 9.37227117888218451020526348936, 9.963855970018970030265382782249, 11.243965593516015509316464796487, 12.29535638150461573830771616149, 13.169213817650933458528747843761, 14.06508484429390729941723635157, 14.674282586424450864453537912121, 15.573465789371760414829994193156, 16.61956878172481077492606927249, 16.95320772548779876540025489895, 18.14263412422489992544869918766, 18.67462173684180124880436864700, 19.5160681479416275974126436630, 20.18786291186045043650770690735, 20.92926679116217684849775058294, 21.57102861807976203465531566481

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