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Properties

Label 1-847-847.514-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.895 - 0.445i$
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  + (0.398 − 0.917i)2-s + (−0.913 − 0.406i)3-s + (−0.683 − 0.730i)4-s + (−0.997 + 0.0760i)5-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (0.669 + 0.743i)9-s + (−0.327 + 0.945i)10-s + (0.327 + 0.945i)12-s + (0.516 + 0.856i)13-s + (0.941 + 0.336i)15-s + (−0.0665 + 0.997i)16-s + (0.161 − 0.986i)17-s + (0.948 − 0.318i)18-s + (0.449 − 0.893i)19-s + (0.736 + 0.676i)20-s + ⋯
L(s)  = 1  + (0.398 − 0.917i)2-s + (−0.913 − 0.406i)3-s + (−0.683 − 0.730i)4-s + (−0.997 + 0.0760i)5-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (0.669 + 0.743i)9-s + (−0.327 + 0.945i)10-s + (0.327 + 0.945i)12-s + (0.516 + 0.856i)13-s + (0.941 + 0.336i)15-s + (−0.0665 + 0.997i)16-s + (0.161 − 0.986i)17-s + (0.948 − 0.318i)18-s + (0.449 − 0.893i)19-s + (0.736 + 0.676i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1836596329 - 0.7814310385i\)
\(L(\frac12)\) \(\approx\) \(0.1836596329 - 0.7814310385i\)
\(L(1)\) \(\approx\) \(0.5850667235 - 0.4892264823i\)
\(L(1)\) \(\approx\) \(0.5850667235 - 0.4892264823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.398 - 0.917i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (-0.997 + 0.0760i)T \)
13 \( 1 + (0.516 + 0.856i)T \)
17 \( 1 + (0.161 - 0.986i)T \)
19 \( 1 + (0.449 - 0.893i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.362 + 0.931i)T \)
31 \( 1 + (-0.797 - 0.603i)T \)
37 \( 1 + (-0.905 - 0.424i)T \)
41 \( 1 + (-0.870 - 0.491i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (0.948 + 0.318i)T \)
53 \( 1 + (-0.0665 - 0.997i)T \)
59 \( 1 + (-0.861 - 0.508i)T \)
61 \( 1 + (-0.595 + 0.803i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (0.999 - 0.0380i)T \)
79 \( 1 + (0.851 + 0.524i)T \)
83 \( 1 + (-0.985 + 0.170i)T \)
89 \( 1 + (-0.0475 - 0.998i)T \)
97 \( 1 + (0.564 - 0.825i)T \)
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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

−22.778393706745408223031750218837, −21.94091805306053592369845108550, −21.0517088503219493493459653536, −20.29948709956933572821414552168, −18.94531769169843713120411082081, −18.33566730497385151926648580363, −17.30794186380526297178487120224, −16.77589178326541267856344681344, −15.93177306130658186364436113670, −15.36318565481809493188970361030, −14.78794724219059498335869402897, −13.570565830006254935170894272, −12.44255788343426459628638559331, −12.23848184017324887605151719564, −11.03254745428180587691986247776, −10.29639776373774815351452643590, −9.041274219809348312888521886421, −8.17325044601265443376875777985, −7.38804087150351349442659627402, −6.42310589812729981977373447344, −5.62556600220137300857304498909, −4.81556009677918126626347090914, −3.88210911568240824959428502480, −3.280602479642222208022007800673, −0.975624230415835239073864051948, 0.49328394342152706534529111155, 1.53770250773910984127033504144, 2.837555898593594502315248219608, 3.86449858300132893295711476503, 4.7827323656145859564224048911, 5.44230650204110528576609492438, 6.73402771761165764784865333558, 7.353814802508890772780739778366, 8.71225719779381188613607823490, 9.52891688639849892062903382537, 10.88652589629952847012530870113, 11.14097697025061902229000650028, 11.937982763298124355323592561843, 12.55448919038583901330976638101, 13.491654304817252194541713056, 14.22961568603589519923872418623, 15.4361260465637831637119432280, 16.0633278026859741446972524595, 17.07198897457424544500464364479, 18.11493935906356339542681500372, 18.68378326275918334974083105266, 19.31919731947235768783299232323, 20.139411290506235558080240194067, 21.030136007087009852099657822658, 21.89295146518272594238320695978

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