↑

Properties

Label 1-1045-1045.68-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.987 + 0.160i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.994 − 0.104i)2-s + (−0.207 − 0.978i)3-s + (0.978 + 0.207i)4-s + (0.104 + 0.994i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s − i·12-s + (−0.406 − 0.913i)13-s + (0.978 − 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.406 + 0.913i)17-s + (0.951 − 0.309i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (−0.104 + 0.994i)24-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.207 − 0.978i)3-s + (0.978 + 0.207i)4-s + (0.104 + 0.994i)6-s + (−0.951 + 0.309i)7-s + (−0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s − i·12-s + (−0.406 − 0.913i)13-s + (0.978 − 0.207i)14-s + (0.913 + 0.406i)16-s + (−0.406 + 0.913i)17-s + (0.951 − 0.309i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (−0.104 + 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4233613340 + 0.03413137450i\)
\(L(\frac12)\) \(\approx\) \(0.4233613340 + 0.03413137450i\)
\(L(1)\) \(\approx\) \(0.4787115855 - 0.1306876469i\)
\(L(1)\) \(\approx\) \(0.4787115855 - 0.1306876469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (-0.207 - 0.978i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (-0.406 + 0.913i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (0.406 + 0.913i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (-0.743 - 0.669i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (0.587 + 0.809i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.994 + 0.104i)T \)
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

−21.594314959963964828593031885073, −20.37989473053814229992395087550, −20.11016120819470281313705996039, −19.20958122699386489196531452315, −18.37716561902345832696616028060, −17.48649951227193204515955745443, −16.6732076697480321761871223710, −16.22330530325275257469651189862, −15.61921169418630066675247468581, −14.67709263512391100785080071387, −13.862734133038921321660474710910, −12.56487540120023010111280471542, −11.61969098688074765613836404188, −11.02342047752705900900128596954, −10.07960557798076134626707339616, −9.428185824899076343292482929904, −9.0658723495370340028907301960, −7.79523102385200782036658072762, −6.88142408688575540818356606715, −6.136205452462908992845765612514, −5.141552800044402902548554548006, −3.96374360832067563934951441584, −3.08274295430637136540421945765, −2.01473868893823947356455143712, −0.35991374206730381726405832490, 0.746986972629847569445259126698, 2.06595482439861307678838265360, 2.67947115220338463208195778895, 3.819367514479818009521766418355, 5.71634510497525252646320873556, 6.0846167890699924455662559235, 7.09629542904881128004094855492, 7.78116508102342505647105400503, 8.586907662528399981452534089669, 9.45185869753204105894536328034, 10.35634140831587773551927963637, 11.11989369424986198998279008062, 12.10015767551280166067635754892, 12.704412071741389977681293147181, 13.294369311649545743163072869667, 14.6896540977373195947120124021, 15.385926585256182705035538907716, 16.44855289166509916300749174594, 16.9536270608351824859454334052, 17.86121726980931250259500822011, 18.43214906630004968793388561534, 19.11798305949335108233365268004, 19.94985834323309199927012782191, 20.18385047382855077767565141619, 21.62898512499821823368707496236

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