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Properties

Label 1-60e2-3600.2131-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.340 + 0.940i$
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 + (−0.406 + 0.913i)11-s + (0.406 + 0.913i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.104 − 0.994i)23-s + (−0.207 − 0.978i)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 + (0.978 − 0.207i)47-s + (−0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (0.406 + 0.913i)59-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−0.406 + 0.913i)11-s + (0.406 + 0.913i)13-s + (0.309 + 0.951i)17-s + (0.951 − 0.309i)19-s + (−0.104 − 0.994i)23-s + (−0.207 − 0.978i)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 + (0.978 − 0.207i)47-s + (−0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (0.406 + 0.913i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.340 + 0.940i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.340 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.441275969 + 1.011067858i\)
\(L(\frac12)\) \(\approx\) \(1.441275969 + 1.011067858i\)
\(L(1)\) \(\approx\) \(1.017162763 + 0.06576405954i\)
\(L(1)\) \(\approx\) \(1.017162763 + 0.06576405954i\)

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

−18.42622491059990984679634338909, −17.8740224849902685641166583811, −17.00423908068473253347268088951, −15.95546467101212090991197827580, −15.892595470608689748275102075145, −15.15678491945954715063335297527, −13.95192344746456543701360498479, −13.79060277899926455641404450876, −12.70964738729716134240167651688, −12.27785327104988878192465816003, −11.37938100798492807874507337895, −10.83726324204311148610855708301, −9.83601533882266781942020637342, −9.35003094412229883027385754057, −8.49913305223206025542459066262, −7.86660065643782356999778716441, −7.0742791598410045337843099177, −6.06560269708417660058063089886, −5.51602256128661602549636147325, −5.019116295368295685818548120054, −3.529161467848048476862085000221, −3.20980741859608709879885242245, −2.38410909849453054604690382620, −1.19145621877322268977875911857, −0.35683918562345170397812746023, 0.74945530844202729846944029279, 1.621619711803989101744376308696, 2.556003469505314139443729934494, 3.49658585673635272611602525678, 4.25707162006486537614707476716, 4.82060549396509831497209772056, 5.952365524653384635697392091244, 6.623812704084133504118335252187, 7.270156431490609407933514132154, 7.99851740412068621141919310568, 8.837574834925800454001565158022, 9.71497143048010083004081823033, 10.23267165467869269799684384181, 10.83897124376346116887560117143, 11.87396400302693595650759329333, 12.32852457264178183220167894154, 13.33352044549997352237850096677, 13.6646287788412886261470972860, 14.49226259934402044256700680980, 15.2969854829779750938287244164, 15.917131654642387436348314241636, 16.67586141088581623298754037010, 17.21093205514586217593156914314, 17.90253675610191883066943679843, 18.78323697831633667796702427402

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