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Properties

Label 1-1800-1800.61-r0-0-0
Degree $1$
Conductor $1800$
Sign $0.709 - 0.704i$
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  + (−0.5 − 0.866i)7-s + (0.978 − 0.207i)11-s + (0.978 + 0.207i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)23-s + (0.104 − 0.994i)29-s + (−0.104 − 0.994i)31-s + (−0.309 − 0.951i)37-s + (−0.978 − 0.207i)41-s + (0.5 + 0.866i)43-s + (−0.104 + 0.994i)47-s + (−0.5 + 0.866i)49-s + (0.809 + 0.587i)53-s + (0.978 + 0.207i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (0.978 − 0.207i)11-s + (0.978 + 0.207i)13-s + (−0.809 + 0.587i)17-s + (0.809 − 0.587i)19-s + (0.669 + 0.743i)23-s + (0.104 − 0.994i)29-s + (−0.104 − 0.994i)31-s + (−0.309 − 0.951i)37-s + (−0.978 − 0.207i)41-s + (0.5 + 0.866i)43-s + (−0.104 + 0.994i)47-s + (−0.5 + 0.866i)49-s + (0.809 + 0.587i)53-s + (0.978 + 0.207i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.709 - 0.704i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.539137351 - 0.6343866380i\)
\(L(\frac12)\) \(\approx\) \(1.539137351 - 0.6343866380i\)
\(L(1)\) \(\approx\) \(1.127699537 - 0.1699743109i\)
\(L(1)\) \(\approx\) \(1.127699537 - 0.1699743109i\)

Euler product

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

−20.25752363594922752677308091351, −19.612082058553954488735293279848, −18.622124595646971920145520300579, −18.29613199905518147687771871411, −17.39980757799722454265490485766, −16.44237815116072134325203859774, −15.932001833250789806174999127509, −15.159905991151876446632358064082, −14.4020741740067392764256462241, −13.583810197882318149363251218062, −12.82369023682529745363158632390, −12.00412160776631238253988154579, −11.48496061493442045536301064627, −10.47809918042893482242750317521, −9.68239842649877417420332004547, −8.69423205369664277296292802913, −8.58650345226158927303673443787, −6.98040607544159539397299492847, −6.66357328001665670393378487629, −5.61934949503605124453441773726, −4.917513576346289548671611848588, −3.72830559648413643917371797977, −3.10801576773909111983329432058, −2.019374076790419151817793003289, −1.01733560225682206800048157433, 0.73744058593470975396583039229, 1.61264329114755214738037366081, 2.889581872902175074936023011059, 3.86499588969590591584186435344, 4.247765330810104929058228048648, 5.57464593982132811001956827259, 6.383108085240509954103449341027, 7.00439883492228290996184525632, 7.86660299114804721139394437778, 8.92092899464345415083087547062, 9.419091657788050079077242302203, 10.359592510099212814932439633387, 11.24064416053585340558720759981, 11.62720327147367389195956872162, 12.89967562464647706867725988834, 13.42837748601733022970079643077, 14.00609530225783189799036561777, 14.959426222718475767838416289724, 15.78760257934707870811177856304, 16.38432946995096271079713146563, 17.29713850195669673836179948371, 17.64438622185498081888526876871, 18.81250157415109797608276443937, 19.42902850329475195706536498985, 20.01368266149448903103643429306

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