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Properties

Label 1-847-847.179-r0-0-0
Degree $1$
Conductor $847$
Sign $0.989 - 0.145i$
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  + (−0.999 + 0.0190i)2-s + (0.913 + 0.406i)3-s + (0.999 − 0.0380i)4-s + (−0.935 + 0.353i)5-s + (−0.921 − 0.389i)6-s + (−0.998 + 0.0570i)8-s + (0.669 + 0.743i)9-s + (0.928 − 0.371i)10-s + (0.928 + 0.371i)12-s + (−0.985 − 0.170i)13-s + (−0.998 − 0.0570i)15-s + (0.997 − 0.0760i)16-s + (−0.398 − 0.917i)17-s + (−0.683 − 0.730i)18-s + (0.861 − 0.508i)19-s + (−0.921 + 0.389i)20-s + ⋯
L(s)  = 1  + (−0.999 + 0.0190i)2-s + (0.913 + 0.406i)3-s + (0.999 − 0.0380i)4-s + (−0.935 + 0.353i)5-s + (−0.921 − 0.389i)6-s + (−0.998 + 0.0570i)8-s + (0.669 + 0.743i)9-s + (0.928 − 0.371i)10-s + (0.928 + 0.371i)12-s + (−0.985 − 0.170i)13-s + (−0.998 − 0.0570i)15-s + (0.997 − 0.0760i)16-s + (−0.398 − 0.917i)17-s + (−0.683 − 0.730i)18-s + (0.861 − 0.508i)19-s + (−0.921 + 0.389i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.989 - 0.145i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.989 - 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9922686298 - 0.07236164618i\)
\(L(\frac12)\) \(\approx\) \(0.9922686298 - 0.07236164618i\)
\(L(1)\) \(\approx\) \(0.8105497635 + 0.06833078450i\)
\(L(1)\) \(\approx\) \(0.8105497635 + 0.06833078450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.999 + 0.0190i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
5 \( 1 + (-0.935 + 0.353i)T \)
13 \( 1 + (-0.985 - 0.170i)T \)
17 \( 1 + (-0.398 - 0.917i)T \)
19 \( 1 + (0.861 - 0.508i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (0.198 - 0.980i)T \)
31 \( 1 + (-0.0665 - 0.997i)T \)
37 \( 1 + (0.272 + 0.962i)T \)
41 \( 1 + (0.0855 - 0.996i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.683 + 0.730i)T \)
53 \( 1 + (0.997 + 0.0760i)T \)
59 \( 1 + (0.820 - 0.572i)T \)
61 \( 1 + (0.483 + 0.875i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (-0.736 + 0.676i)T \)
73 \( 1 + (-0.179 - 0.983i)T \)
79 \( 1 + (0.964 + 0.263i)T \)
83 \( 1 + (-0.0285 - 0.999i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (0.774 - 0.633i)T \)
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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

−21.79907520542294895014184255933, −21.06812021821739138680404240498, −20.01413388062163981594852585424, −19.75492328142169372548382539436, −19.158085121674487076495403330116, −18.2508491386852700544348644962, −17.50948144589069172509339429156, −16.473909550726226001138365156650, −15.77969404597085081926622026787, −14.97590751792102177136611221790, −14.34655955330760757953100197077, −12.96896811702229634271089435745, −12.2913791718902912759911971216, −11.5773646521468362248727497502, −10.48497952734181621206767999337, −9.52815707094421284396194447817, −8.82526345377870558749654781816, −8.063677724052456563677764414544, −7.38775722243106216542587371112, −6.76091318419778635562378647501, −5.31087911620372838792313388901, −3.8933703871942351592147210989, −3.12445486560691536019536247989, −1.988687281818135708691584558936, −1.01893161584403991584770482341, 0.676564137772373847607205136577, 2.44116439460093746359541517005, 2.8171749877934794362786664663, 4.03646864553081762742497236787, 5.04996510632812671869548045623, 6.64221019059375530725318576606, 7.45593826979804273207985881247, 7.94018084928794798876450065748, 8.908467827152048360818135728457, 9.64343516620597062624647353652, 10.39708748909638613901843809197, 11.35107917413962107856262842444, 12.02045720231134257254802490074, 13.19524669547897133815789951165, 14.406583282458010231032341227433, 14.99403091137643634115476479327, 15.76600666438627141380243212883, 16.27598119086619396089655458527, 17.33172523523287427788886121269, 18.33979067403386718752510502034, 19.03553215410815841421804583517, 19.61796837604388476025055191198, 20.35588963088868674082519043031, 20.83258658300550566277594989357, 22.08870413803730243845745502340

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