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Properties

Label 1-1800-1800.1219-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.282 - 0.959i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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 + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)13-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.913 + 0.406i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (−0.809 − 0.587i)37-s + (−0.104 + 0.994i)41-s + (0.5 − 0.866i)43-s + (0.669 + 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)13-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (0.913 + 0.406i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (−0.809 − 0.587i)37-s + (−0.104 + 0.994i)41-s + (0.5 − 0.866i)43-s + (0.669 + 0.743i)47-s + (−0.5 − 0.866i)49-s + (0.309 − 0.951i)53-s + (−0.104 + 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.282 - 0.959i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.282 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3540529139 - 0.4732722982i\)
\(L(\frac12)\) \(\approx\) \(0.3540529139 - 0.4732722982i\)
\(L(1)\) \(\approx\) \(0.8742307252 + 0.06744344257i\)
\(L(1)\) \(\approx\) \(0.8742307252 + 0.06744344257i\)

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

−20.23468750796535175717849150104, −19.66319932657054996298134249947, −18.798388982314729493433310416136, −17.90645171361065086417101151916, −17.22244280344671571511742569230, −16.77478300977463063440887679638, −15.595938628866727091824305562757, −15.21559287800204450533074757562, −14.34892152420561235924613437549, −13.33860368358665735591661334994, −12.88320717044154662259120672155, −12.23211623287378157814344121117, −10.8968580220518896456943989764, −10.63729558137609503217070118066, −9.68309631154789937561162106836, −8.99984951143563888966156418122, −7.91724991857808365767602134401, −7.20697994134768084190684624723, −6.61595707206157722311446492276, −5.50496648605296884898043940608, −4.706906621230149936436212306626, −3.81665261038906786358576779364, −3.0024936794818655039896018687, −1.94915294220821077157367260926, −0.83178254071812356001175714332, 0.12746487409891472098157187687, 1.39417850110327477222244558459, 2.451071485911319457785549805804, 3.230333155516738615726511825789, 4.11224019635080070185685289495, 5.30736596489904302309967517886, 5.79778174108492513345031457340, 6.76821765988016239496884342369, 7.49707782986755052482745629366, 8.65917128876760440099615280928, 9.09578373203160028866478825690, 9.84943497425305173462707273235, 10.914322049924621524156786411205, 11.626597421804122817980060752313, 12.21113029941409905592004506171, 13.17004415350549736281807539336, 13.829136769074838575925443228822, 14.58788494103986380440518310995, 15.44399357846080050386328790838, 16.27698654606728734338892816370, 16.543466452451375759171547751278, 17.691421580879325670041346198895, 18.5053417924121257350414094462, 19.02998839743000583301740727013, 19.54361782550255373327423342814

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