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Properties

Label 1-1800-1800.1067-r1-0-0
Degree $1$
Conductor $1800$
Sign $-0.959 + 0.282i$
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  + (0.866 + 0.5i)7-s + (0.104 + 0.994i)11-s + (−0.994 − 0.104i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.406 − 0.913i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (−0.587 + 0.809i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (−0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.104 + 0.994i)11-s + (−0.994 − 0.104i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (0.406 − 0.913i)23-s + (−0.669 − 0.743i)29-s + (−0.669 + 0.743i)31-s + (−0.587 + 0.809i)37-s + (0.104 − 0.994i)41-s + (0.866 + 0.5i)43-s + (−0.743 + 0.669i)47-s + (0.5 + 0.866i)49-s + (−0.951 − 0.309i)53-s + (−0.104 + 0.994i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.959 + 0.282i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ -0.959 + 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09646409592 + 0.6694144505i\)
\(L(\frac12)\) \(\approx\) \(0.09646409592 + 0.6694144505i\)
\(L(1)\) \(\approx\) \(0.9963053315 + 0.1293904951i\)
\(L(1)\) \(\approx\) \(0.9963053315 + 0.1293904951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.994 - 0.104i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (0.406 - 0.913i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.104 - 0.994i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.104 + 0.994i)T \)
67 \( 1 + (0.743 + 0.669i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (0.587 + 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−19.63795915891754131362181441409, −18.94463227461662045684319991678, −18.299135961113766136295720726995, −17.23579831715911714484814552567, −16.89933999895162409038437594603, −16.15648860337990787472185163541, −15.042675450617810004969312586567, −14.42205211464508986563079245064, −13.971020588106939275169128542779, −12.92200278498177176062532692626, −12.222230791212589483686779577677, −11.29225858428952970754005168351, −10.81844652459212540695948542736, −9.86364486524769343630465121491, −9.122980109026974972606774002706, −8.00500206945744051893304005431, −7.70721137305589450586070460448, −6.68408288443592026951290131596, −5.59316254514869232344902787903, −5.10907234759745571323459631469, −3.92382074682955305065020894687, −3.34694083988633917431251237547, −2.03707793681184559013034986755, −1.26162682446729136168163639776, −0.1198147962112678112529047198, 1.18242551532233775309464952902, 2.18991819116394229043579579678, 2.8415235047878889619821854539, 4.18621265489692030399927881285, 4.918572221005869702788495274453, 5.475322045239232401138889565415, 6.71061606326115314560570571105, 7.400746388951366934973046605067, 8.127119457976757570645384260, 9.08059426305486129171769971635, 9.71762445885593807998968657556, 10.62154100608486758736905508879, 11.40358413939689370803633781534, 12.28508297959659829902311440498, 12.61801373986482230012582770260, 13.79210291043536233289703040677, 14.68083835374295402270859869297, 14.93554368581356664826296258416, 15.82845675740781405129388560544, 16.837007758393416816445997517416, 17.46999585647503874085778952093, 18.01773122201554723061525967103, 18.91946769263984269811519270067, 19.53789006738545293074078454366, 20.546334898230722554242230846486

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