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Properties

Label 1-1045-1045.292-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

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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: $\chi_{1045} (292, \cdot )$
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 \( 1 + (0.994 - 0.104i)T \)
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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

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

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