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Properties

Label 1-1800-1800.797-r0-0-0
Degree $1$
Conductor $1800$
Sign $-0.916 + 0.400i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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  + (−0.866 − 0.5i)7-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.994 − 0.104i)23-s + (0.978 + 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (−0.866 − 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (−0.913 − 0.406i)59-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s + (0.913 − 0.406i)11-s + (−0.406 + 0.913i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (−0.994 − 0.104i)23-s + (0.978 + 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.587 + 0.809i)37-s + (−0.913 − 0.406i)41-s + (−0.866 − 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.5 + 0.866i)49-s + (0.951 − 0.309i)53-s + (−0.913 − 0.406i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.916 + 0.400i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ -0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01821360798 + 0.08718362399i\)
\(L(\frac12)\) \(\approx\) \(0.01821360798 + 0.08718362399i\)
\(L(1)\) \(\approx\) \(0.7768512248 - 0.04057857096i\)
\(L(1)\) \(\approx\) \(0.7768512248 - 0.04057857096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.994 - 0.104i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.743 - 0.669i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−19.97236623115307685855769066160, −19.30043608022803010809077455372, −18.23700527230334048320940291235, −17.86588342757115182376955178126, −16.7932503301040042388298746879, −16.30041967375302641937854273012, −15.28473289309303850360963298766, −14.92514348035875340397341507731, −13.89011071756739865593758560655, −13.13653814336921165837567160839, −12.27827380057505682597795897765, −11.94189578557000306750857650480, −10.76964435490155694283807959341, −9.963386998108973456465548981671, −9.40985431488211004007568394528, −8.52189210079761378516233626984, −7.70854429024353612977880870246, −6.70112867863060703467247783007, −6.11197134574765969413198205528, −5.275923721086239778285965910127, −4.1688125644052255920238933643, −3.44580727836090186451572894791, −2.46682915415177816566459204601, −1.54945204285917583171935493470, −0.031367621228065306024729233575, 1.26828126876440409836568455908, 2.38676243395002914970236697474, 3.34025837824566655591982472881, 4.17049968899917446178297714866, 4.89749339374391682989505597670, 6.19070376484107263608869170034, 6.7057311575465062681584268673, 7.33674126806694326900945073517, 8.58821497131260930588988180623, 9.20735228253521492446660698547, 9.87649007984227046767674254103, 10.77836686492040468025498735112, 11.63362549928744006564165945932, 12.188758653495869856234995688181, 13.28391316178291896840935777368, 13.75877911677391204759327410453, 14.471495034197165472237325638056, 15.46797303921547229625921334277, 16.19830460733234878184453669072, 16.7538210511278644365060612166, 17.49748786778897665866618397664, 18.34957828263001729622130698515, 19.16901768775666979523176864875, 20.00052986313866687513969422657, 20.04262028328982029032844459784

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