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Properties

Label 1-1800-1800.221-r1-0-0
Degree $1$
Conductor $1800$
Sign $0.916 - 0.400i$
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 + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.104 − 0.994i)23-s + (−0.978 − 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (0.5 − 0.866i)43-s + (0.978 + 0.207i)47-s + (−0.5 − 0.866i)49-s + (0.309 + 0.951i)53-s + (0.913 + 0.406i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)13-s + (−0.309 + 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.104 − 0.994i)23-s + (−0.978 − 0.207i)29-s + (−0.978 + 0.207i)31-s + (0.809 − 0.587i)37-s + (−0.913 − 0.406i)41-s + (0.5 − 0.866i)43-s + (0.978 + 0.207i)47-s + (−0.5 − 0.866i)49-s + (0.309 + 0.951i)53-s + (0.913 + 0.406i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.916 - 0.400i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ 0.916 - 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.371509772 - 0.2865233194i\)
\(L(\frac12)\) \(\approx\) \(1.371509772 - 0.2865233194i\)
\(L(1)\) \(\approx\) \(0.9282012241 + 0.05246123687i\)
\(L(1)\) \(\approx\) \(0.9282012241 + 0.05246123687i\)

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

−20.03701118539361861142286007275, −19.48828345462815422539454119423, −18.68170242829597667378944602126, −17.683067695747060990096998911638, −17.08590105153443170097134591851, −16.53748426750031307119306943559, −15.65829337793271638226855561799, −14.79064224801827091056212734114, −14.182039784224442731807562332658, −13.29168535687287726297904712190, −12.79089497468301830847315199052, −11.59878794271173843887175141295, −11.31111938119530051964327531238, −10.098426448846171497753854998833, −9.49541474749035536911691565443, −8.95930571169790069331641046562, −7.567133414327229746684614862459, −7.110342890319299030921729144804, −6.46424692593414536175753486335, −5.26702592376760014174084072160, −4.45298447626281933394580309058, −3.72466160832734678633786891974, −2.72551133050786207113763004708, −1.69998373187906382760022937569, −0.60868846638980392177370969136, 0.38080053422250438056472853821, 1.74291510410655606660699438975, 2.508648557485331960410491295, 3.56321744345926224407598588306, 4.26516272892141665082835244199, 5.52726781234143063147906000299, 5.99484204298713626501365948124, 6.87549693795945713643951394939, 7.8116508432779417039616201885, 8.773384471138224966188152994931, 9.21243478397929197912519700853, 10.21289266189159932014294718413, 10.88258165285512775251029496565, 11.97579644910035748624086103616, 12.424203139275136515154370289767, 13.09555749065942393246655900236, 14.20273060327270638545996923191, 14.867121113719373069748459067760, 15.34230829432505140834176989658, 16.53609533968631611723947288735, 16.81566001528356257273070561521, 17.76287821461773557957636706378, 18.68821338189266734163310204623, 19.13791134422676572577350835018, 19.895222187073279649272566906377

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