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Properties

Label 1-847-847.530-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.0643 - 0.997i$
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  + (−0.380 + 0.924i)2-s + (−0.669 − 0.743i)3-s + (−0.710 − 0.703i)4-s + (0.905 − 0.424i)5-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (−0.104 + 0.994i)9-s + (0.0475 + 0.998i)10-s + (−0.0475 + 0.998i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)15-s + (0.00951 + 0.999i)16-s + (−0.969 + 0.244i)17-s + (−0.879 − 0.475i)18-s + (−0.0665 − 0.997i)19-s + (−0.941 − 0.336i)20-s + ⋯
L(s)  = 1  + (−0.380 + 0.924i)2-s + (−0.669 − 0.743i)3-s + (−0.710 − 0.703i)4-s + (0.905 − 0.424i)5-s + (0.941 − 0.336i)6-s + (0.921 − 0.389i)8-s + (−0.104 + 0.994i)9-s + (0.0475 + 0.998i)10-s + (−0.0475 + 0.998i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)15-s + (0.00951 + 0.999i)16-s + (−0.969 + 0.244i)17-s + (−0.879 − 0.475i)18-s + (−0.0665 − 0.997i)19-s + (−0.941 − 0.336i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0643 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0643 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.0643 - 0.997i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.0643 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4883534671 - 0.5208811632i\)
\(L(\frac12)\) \(\approx\) \(0.4883534671 - 0.5208811632i\)
\(L(1)\) \(\approx\) \(0.7026909568 - 0.07160109075i\)
\(L(1)\) \(\approx\) \(0.7026909568 - 0.07160109075i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.380 + 0.924i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
5 \( 1 + (0.905 - 0.424i)T \)
13 \( 1 + (-0.362 - 0.931i)T \)
17 \( 1 + (-0.969 + 0.244i)T \)
19 \( 1 + (-0.0665 - 0.997i)T \)
23 \( 1 + (0.580 - 0.814i)T \)
29 \( 1 + (0.985 - 0.170i)T \)
31 \( 1 + (-0.548 + 0.836i)T \)
37 \( 1 + (0.161 - 0.986i)T \)
41 \( 1 + (-0.564 + 0.825i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.879 + 0.475i)T \)
53 \( 1 + (0.00951 - 0.999i)T \)
59 \( 1 + (-0.997 + 0.0760i)T \)
61 \( 1 + (-0.991 - 0.132i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.217 - 0.976i)T \)
79 \( 1 + (0.683 + 0.730i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (0.786 - 0.618i)T \)
97 \( 1 + (-0.0855 - 0.996i)T \)
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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

−21.92086387287788543724904145352, −21.67801604139906754151934507598, −20.82450698155679391953615975344, −20.16728250174399108205396010149, −18.957393069266574085931585486114, −18.386164139094661262424454118916, −17.41628143488108774332788533880, −17.07687859684881608226752655608, −16.172200553435015871164929389944, −15.0360488232973065562157506394, −14.092009616010334894125781495591, −13.35713051336371691364062457228, −12.27665388310353947900422020689, −11.54084361162226235855391270799, −10.78714464537624617855674170478, −10.081739786776379680192393272111, −9.39938007325335494479305031170, −8.757476429197764434640419381976, −7.27006828522753265531124923171, −6.32404124116196762669830555324, −5.26103065284373418892080052078, −4.41688225844967384375221981845, −3.453195643483746007100979730941, −2.381047840766016426417656012347, −1.3638522375897640386283340508, 0.425903618628270189012393416854, 1.5021162239341696163134491147, 2.63764152088862812600677318109, 4.73245651404615737091279429555, 5.07913696607737713764258398479, 6.22794796507888400825347909706, 6.613594466908359827846181259537, 7.6745057805719928487221557124, 8.548133436354418292009313589042, 9.34544841999670946445939360284, 10.42721307127497424779527774745, 11.00547825573433037150934394902, 12.52957108009769882867014905811, 13.027711089976392659697655796950, 13.75324711805677592665172821002, 14.65230594449246572375819376511, 15.674452797958098247788048236221, 16.51392338125251617098605546349, 17.213349421992705226756334616219, 17.96021852001147889204882799190, 18.08656839227895793880081474234, 19.49152959534737527552664901610, 19.898756343001615065155168987770, 21.3493645902663073828606644815, 22.18032024224487826992413942939

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