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Properties

Label 1-847-847.160-r0-0-0
Degree $1$
Conductor $847$
Sign $0.0207 - 0.999i$
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  + (−0.198 − 0.980i)2-s + (−0.309 − 0.951i)3-s + (−0.921 + 0.389i)4-s + (−0.610 − 0.791i)5-s + (−0.870 + 0.491i)6-s + (0.564 + 0.825i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (0.974 + 0.226i)13-s + (−0.564 + 0.825i)15-s + (0.696 − 0.717i)16-s + (−0.362 − 0.931i)17-s + (0.736 + 0.676i)18-s + (0.774 + 0.633i)19-s + (0.870 + 0.491i)20-s + ⋯
L(s)  = 1  + (−0.198 − 0.980i)2-s + (−0.309 − 0.951i)3-s + (−0.921 + 0.389i)4-s + (−0.610 − 0.791i)5-s + (−0.870 + 0.491i)6-s + (0.564 + 0.825i)8-s + (−0.809 + 0.587i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (0.974 + 0.226i)13-s + (−0.564 + 0.825i)15-s + (0.696 − 0.717i)16-s + (−0.362 − 0.931i)17-s + (0.736 + 0.676i)18-s + (0.774 + 0.633i)19-s + (0.870 + 0.491i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0207 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0207 - 0.999i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.0207 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6067888217 - 0.6195250771i\)
\(L(\frac12)\) \(\approx\) \(0.6067888217 - 0.6195250771i\)
\(L(1)\) \(\approx\) \(0.5622236831 - 0.4951034268i\)
\(L(1)\) \(\approx\) \(0.5622236831 - 0.4951034268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.198 - 0.980i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.610 - 0.791i)T \)
13 \( 1 + (0.974 + 0.226i)T \)
17 \( 1 + (-0.362 - 0.931i)T \)
19 \( 1 + (0.774 + 0.633i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (0.254 + 0.967i)T \)
31 \( 1 + (-0.0855 + 0.996i)T \)
37 \( 1 + (0.516 + 0.856i)T \)
41 \( 1 + (0.993 + 0.113i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.736 - 0.676i)T \)
53 \( 1 + (0.696 + 0.717i)T \)
59 \( 1 + (-0.993 + 0.113i)T \)
61 \( 1 + (0.198 - 0.980i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.998 - 0.0570i)T \)
73 \( 1 + (0.897 + 0.441i)T \)
79 \( 1 + (-0.941 + 0.336i)T \)
83 \( 1 + (-0.466 + 0.884i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.610 + 0.791i)T \)
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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

−22.517634642272921699870228680847, −21.84303075440638613520743428328, −20.82071793852699413555917883519, −19.79972995264352396291225860785, −18.991946499110353452448931046185, −18.04075159064908784549513208895, −17.519571478986104236028121096232, −16.45053346026916173925473528511, −15.839185934134863980581838823737, −15.25908758910723682439290343384, −14.586113690049077703777462975479, −13.773075248460422855059096133363, −12.654863923048404999578077034096, −11.370392282930934410530580558610, −10.76991408357220994202421369011, −9.97000861337643849870384949009, −9.0051268786236721426092334178, −8.224730320928269591656933877136, −7.31648373503095941060211551212, −6.1953215947617100302061737077, −5.78704801265308247409867631149, −4.32267128978204356835327486879, −4.00830349656040872413405735940, −2.74702263175130425240985843210, −0.6298588491129687410681922423, 0.9349269857129595209034189123, 1.53212328521093650595277375652, 2.861063892797892473596297389490, 3.82134322713849022097353551862, 4.9309550704552002249496224108, 5.71463304243657505912966683471, 7.1353242908134429847717488828, 7.9078172208503926682305849504, 8.71929153302344360303493895150, 9.42049184123535934745750331656, 10.77153198431650022153736462052, 11.457638457386653514325505559201, 12.09907702681750834467398920546, 12.7345811869723749896447436156, 13.632247589349663053288613581132, 14.10356218253016225893344832758, 15.73969623907252541574542435499, 16.479079958550833393446264104880, 17.32685089247291570382455418197, 18.250860919350155250361506419140, 18.62030650553046736522642660442, 19.70994795371816352566522463463, 20.08328000298228413787698034197, 20.890158009538363232222038809885, 21.84827990923177024935576790575

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