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Properties

Label 1-60e2-3600.1589-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.859 + 0.510i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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 + (0.743 − 0.669i)11-s + (0.743 + 0.669i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.978 − 0.207i)23-s + (0.406 + 0.913i)29-s + (0.913 + 0.406i)31-s + (0.951 − 0.309i)37-s + (0.669 − 0.743i)41-s + (0.866 + 0.5i)43-s + (0.913 − 0.406i)47-s + (−0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (−0.743 − 0.669i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.743 − 0.669i)11-s + (0.743 + 0.669i)13-s + (−0.809 + 0.587i)17-s + (0.587 + 0.809i)19-s + (0.978 − 0.207i)23-s + (0.406 + 0.913i)29-s + (0.913 + 0.406i)31-s + (0.951 − 0.309i)37-s + (0.669 − 0.743i)41-s + (0.866 + 0.5i)43-s + (0.913 − 0.406i)47-s + (−0.5 − 0.866i)49-s + (0.587 − 0.809i)53-s + (−0.743 − 0.669i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.859 + 0.510i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.859 + 0.510i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.798299167 + 0.7681533871i\)
\(L(\frac12)\) \(\approx\) \(2.798299167 + 0.7681533871i\)
\(L(1)\) \(\approx\) \(1.226102506 + 0.1719172534i\)
\(L(1)\) \(\approx\) \(1.226102506 + 0.1719172534i\)

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 \( 1 + (0.743 - 0.669i)T \)
13 \( 1 + (0.743 + 0.669i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.406 + 0.913i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (-0.743 - 0.669i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−18.357913457764986453895123502123, −17.67815092104687120482315171727, −17.17721554423236290438974601225, −16.46754644342080935218152249378, −15.517258020395051516066750061803, −15.32147818240262440626326194410, −14.17759946667799000539100393603, −13.539068606567565209249059491729, −13.12827461288049874171712977629, −12.25133300744530723320225242392, −11.37830068201561717098671831200, −10.899285190925751365024938222594, −9.97720542722866372386581104256, −9.41691871974335243970566248850, −8.73179155256868246487014170223, −7.68207047521362541588932314582, −7.12967413388387085297092045126, −6.44071782722760469858473959518, −5.69475846108382208699953479508, −4.47129139888122543711647615271, −4.23915499643541727982567366902, −3.08474366616227561438543557625, −2.50400695082271477623499034928, −1.05893955712698542959097312367, −0.73918117610656102193571507251, 0.690980235396998313389328759, 1.54537296817699868959871306434, 2.501545778361979807152426254834, 3.3537278463021232774997650951, 4.012256547417702964345315335703, 4.96360709889188967570422682650, 5.95828478542419218352770066849, 6.30947458354080146467754424198, 7.093614953793537376624556524453, 8.18733306270362116324659277271, 8.96202501874143667204166912373, 9.12662041490624890629281499626, 10.22650883067997359323328620628, 11.0224466925813650985573964141, 11.620868658104020150935519507523, 12.354502629468242616596769111041, 12.99294977416982769935110963275, 13.80972979993560660819203926480, 14.42566740443205782318638578278, 15.1599140354987194094701687196, 16.0332623951020659832439285736, 16.3045623528400864472299141743, 17.200568325910327071462441995259, 17.96797345834227871031894518053, 18.6817154011471805571157551754

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