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Properties

Label 1-60e2-3600.211-r1-0-0
Degree $1$
Conductor $3600$
Sign $-0.246 - 0.969i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.246 - 0.969i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ -0.246 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3810779581 + 0.4903984757i\)
\(L(\frac12)\) \(\approx\) \(-0.3810779581 + 0.4903984757i\)
\(L(1)\) \(\approx\) \(0.7503423603 + 0.3263114061i\)
\(L(1)\) \(\approx\) \(0.7503423603 + 0.3263114061i\)

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

−18.0271217058478121853525925639, −17.46694824820660200706819649686, −16.4963398135962262080990964451, −15.94530249419629260658334944161, −15.576076024075539293856265347700, −14.49422562613750113709804637824, −13.66573966200784921661361905137, −13.314497274130135997970381345736, −12.75058279101864726236219520894, −11.49631214503793942492604933196, −11.185539433487278260580851402799, −10.24046704344825673597695892807, −9.83208846418941872907778015627, −8.78713023329261365445034813093, −8.153672786783160663086161211754, −7.40214069285679272817067399044, −6.62986921632686310101715596520, −5.9443363142755328106635041534, −5.077566317090613995244401175457, −4.28687452806810470101448193558, −3.35877590119021803800386320683, −2.85889398859588825086016534154, −1.71692443537226695542764307481, −0.54109822245214877096767215297, −0.14697263982057535286615496567, 1.45042432647107172762464071208, 2.065834112347686451184026744063, 2.982447320791602865623642447179, 3.81110713459159127293464121802, 4.62618796915901682488321509184, 5.55313030552292956446456520725, 6.13926898566046281895650803094, 6.86794263102416721348546422430, 7.80100622987954298160179501762, 8.48206770274369682606272245866, 9.21494237500064202592800669549, 9.88033751485771377204146020016, 10.593773636240183153637960788655, 11.47398089873645414347150581297, 12.07649545287735366625852269454, 12.8661173126667886738197769064, 13.30609245891056270431411695182, 14.266390561259354415151805726104, 14.94594322134066570399911242464, 15.672260148894650927582517542750, 16.15794013084428971782221843829, 16.80804071790623219132405830433, 17.95504323938151105279481139074, 18.28265264563062468998023615805, 18.79262571030444314737087045091

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