↑

Properties

Label 1-847-847.16-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.978 − 0.207i)3-s + (−0.830 + 0.556i)4-s + (−0.625 + 0.780i)5-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (0.913 + 0.406i)9-s + (0.928 + 0.371i)10-s + (0.928 − 0.371i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (0.380 − 0.924i)16-s + (0.749 + 0.662i)17-s + (0.123 − 0.992i)18-s + (−0.398 − 0.917i)19-s + (0.0855 − 0.996i)20-s + ⋯
L(s)  = 1  + (−0.290 − 0.956i)2-s + (−0.978 − 0.207i)3-s + (−0.830 + 0.556i)4-s + (−0.625 + 0.780i)5-s + (0.0855 + 0.996i)6-s + (0.774 + 0.633i)8-s + (0.913 + 0.406i)9-s + (0.928 + 0.371i)10-s + (0.928 − 0.371i)12-s + (−0.466 − 0.884i)13-s + (0.774 − 0.633i)15-s + (0.380 − 0.924i)16-s + (0.749 + 0.662i)17-s + (0.123 − 0.992i)18-s + (−0.398 − 0.917i)19-s + (0.0855 − 0.996i)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: $\chi_{847} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.989 + 0.145i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5085266271 + 0.03708453816i\)
\(L(\frac12)\) \(\approx\) \(0.5085266271 + 0.03708453816i\)
\(L(1)\) \(\approx\) \(0.5209680446 - 0.1499646943i\)
\(L(1)\) \(\approx\) \(0.5209680446 - 0.1499646943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.290 - 0.956i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
5 \( 1 + (-0.625 + 0.780i)T \)
13 \( 1 + (-0.466 - 0.884i)T \)
17 \( 1 + (0.749 + 0.662i)T \)
19 \( 1 + (-0.398 - 0.917i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (-0.736 - 0.676i)T \)
31 \( 1 + (-0.969 + 0.244i)T \)
37 \( 1 + (0.345 + 0.938i)T \)
41 \( 1 + (0.516 - 0.856i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.123 + 0.992i)T \)
53 \( 1 + (0.380 + 0.924i)T \)
59 \( 1 + (-0.999 + 0.0190i)T \)
61 \( 1 + (-0.683 - 0.730i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.870 + 0.491i)T \)
73 \( 1 + (-0.432 - 0.901i)T \)
79 \( 1 + (0.548 + 0.836i)T \)
83 \( 1 + (0.941 + 0.336i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (-0.362 + 0.931i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.37692772146131481083686923198, −21.43580004515221444370589851054, −20.569526530641331496189894939512, −19.43698896636749651595991277536, −18.661497397342480355701623258524, −18.02390405889046917185054482257, −16.81541895411952980590115739808, −16.5786076245395061327478562358, −16.08781710211920904612405015445, −14.95620227462826902688830148864, −14.36276250752654458602994270387, −12.99992136163951112506134956159, −12.42095152826554135368506681320, −11.50028367653902766201873696420, −10.535346077690891240117642961853, −9.53848208977269276983734375358, −8.92880719622673450889455802987, −7.73806199627373030339681451973, −7.1420374483657133142717567894, −6.082868209398673483813284688015, −5.284007231642802418180176554890, −4.52783315849233068544400578395, −3.79978587322461521011020348440, −1.6070758518833942602072181480, −0.43574054877832256128203673007, 0.82808340196640346186451298307, 2.1284475718184963367177663207, 3.2173945930484853730256732859, 4.10897940030556641212901013139, 5.13467949708790718785219976867, 6.1043561268386396322334351665, 7.46582961737502893219745987408, 7.73198912312390322899671465071, 9.16665020574067897835949806171, 10.17723761119471395518701762315, 10.80699607623212151774679863084, 11.36742161278582937734230225172, 12.25993813596651570246836847971, 12.8163533959309083434849219592, 13.7962439164934999211640105309, 14.96303594244828448933183401824, 15.7155137642015845354882982460, 16.94928222809004525259452042113, 17.435614299070543009951636028157, 18.24354315117853265767156179858, 19.00985017921697267978343741023, 19.49515136085929288116979855855, 20.46521895648556256027186451807, 21.59137034301624236908892923038, 22.04702617329776982430614247863

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