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Properties

Label 1-1800-1800.709-r0-0-0
Degree $1$
Conductor $1800$
Sign $0.0906 + 0.995i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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 + (−0.669 + 0.743i)11-s + (0.669 + 0.743i)13-s + (0.809 + 0.587i)17-s + (0.809 + 0.587i)19-s + (0.978 + 0.207i)23-s + (−0.913 − 0.406i)29-s + (0.913 − 0.406i)31-s + (0.309 − 0.951i)37-s + (0.669 + 0.743i)41-s + (−0.5 − 0.866i)43-s + (−0.913 − 0.406i)47-s + (−0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (−0.669 − 0.743i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.669 + 0.743i)11-s + (0.669 + 0.743i)13-s + (0.809 + 0.587i)17-s + (0.809 + 0.587i)19-s + (0.978 + 0.207i)23-s + (−0.913 − 0.406i)29-s + (0.913 − 0.406i)31-s + (0.309 − 0.951i)37-s + (0.669 + 0.743i)41-s + (−0.5 − 0.866i)43-s + (−0.913 − 0.406i)47-s + (−0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (−0.669 − 0.743i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0906 + 0.995i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.0906 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.268132121 + 1.157963752i\)
\(L(\frac12)\) \(\approx\) \(1.268132121 + 1.157963752i\)
\(L(1)\) \(\approx\) \(1.131023892 + 0.3323885431i\)
\(L(1)\) \(\approx\) \(1.131023892 + 0.3323885431i\)

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

−20.190207870205939010303778872766, −19.254522182391502823884299891404, −18.477553286502207170988902298736, −17.88407457179756136899941466073, −17.072906053779772674443548219613, −16.321416782746059721867259874432, −15.70958306497630847780115307372, −14.79423549015120847970645884321, −13.99058920039204858630972431420, −13.39218278548266768882456798703, −12.76435714105363252169522545384, −11.55360284595287184471301360614, −11.057728914010887723957815176576, −10.34851041965403548748973426683, −9.521371871551278529894549897252, −8.49537253277636553200167530272, −7.82923617436388408800716744461, −7.18240272999742346808573221112, −6.14358147379643015879600588966, −5.227186740540634540204132895918, −4.647150423117534416491078382297, −3.30880684368633785512755778707, −2.99458527422166355403277076642, −1.41082153039154377106543245657, −0.67713121831161855335785900222, 1.29722185279290589728061155129, 2.062507648421776595687182389418, 3.04956218514027973654611258701, 4.03144165848024927154909189316, 5.00449189653119420499337491850, 5.65876162985288032336541353648, 6.48895202951914015883470972, 7.63503981217889433047545490306, 8.07377992959942514098613666045, 9.13407561816489032602522169298, 9.68175072729068274009213016283, 10.68817103360483678869556152322, 11.44943781498872043924607852845, 12.1423927556410706155855228164, 12.85953019626252268371750814492, 13.69705539904510666585925278317, 14.594257151680884434315021560760, 15.15242250308890838874615041417, 15.87865859648367864824573968430, 16.69026587581118208395283955319, 17.4807986649282157032924640180, 18.36906595051051430943474109022, 18.68811481074444080146761052614, 19.527677136743249617057437744735, 20.6617846923950145549023063472

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