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Properties

Label 1-1800-1800.1667-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  + (−0.866 + 0.5i)7-s + (−0.913 − 0.406i)11-s + (0.406 + 0.913i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.994 + 0.104i)23-s + (0.978 − 0.207i)29-s + (0.978 + 0.207i)31-s + (−0.587 + 0.809i)37-s + (−0.913 + 0.406i)41-s + (−0.866 + 0.5i)43-s + (−0.207 − 0.978i)47-s + (0.5 − 0.866i)49-s + (−0.951 − 0.309i)53-s + (0.913 − 0.406i)59-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (−0.913 − 0.406i)11-s + (0.406 + 0.913i)13-s + (0.951 − 0.309i)17-s + (−0.309 − 0.951i)19-s + (−0.994 + 0.104i)23-s + (0.978 − 0.207i)29-s + (0.978 + 0.207i)31-s + (−0.587 + 0.809i)37-s + (−0.913 + 0.406i)41-s + (−0.866 + 0.5i)43-s + (−0.207 − 0.978i)47-s + (0.5 − 0.866i)49-s + (−0.951 − 0.309i)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: $\chi_{1800} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ 0.916 + 0.400i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.300753313 + 0.2717415251i\)
\(L(\frac12)\) \(\approx\) \(1.300753313 + 0.2717415251i\)
\(L(1)\) \(\approx\) \(0.8810775913 + 0.04527311871i\)
\(L(1)\) \(\approx\) \(0.8810775913 + 0.04527311871i\)

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

−20.05494805580015173990790192128, −19.16185820912569921863622673696, −18.59188781176012630367796971846, −17.724439365033261344756387517256, −17.06272100802600322243671627429, −16.078794570098947077695337459725, −15.77929170090056242801614876226, −14.79632741731067965693641132850, −13.95273266404462143662399182688, −13.23952591747441128036386630532, −12.50407366404307820702066248008, −11.994578870744981451170666872014, −10.487715314920689811398347478265, −10.38452966792033386041982665760, −9.629816341298675019936452200183, −8.33155539984642131043314398575, −7.91269353411323911850680429294, −6.95288509555939899966033581832, −6.05397049221037100460315892544, −5.424851893718718422191179784676, −4.2994286493996668147886935099, −3.45449451838046254924421725583, −2.74151592330004330175654857568, −1.532236819496252745504098896232, −0.43355444514298301780737075885, 0.51419044558939664801278565878, 1.808397023751797557137431215600, 2.8356909946112896328284375740, 3.41895941743320832317448653436, 4.61761488846610850057423206869, 5.38055734559124583865949065922, 6.37299008967998133064489325384, 6.813105497045304650998909013625, 8.10910467190342233025711555315, 8.551455418196806293087574015934, 9.682179120883050984844287285559, 10.06076386901093193062739334845, 11.13262001539724636688261967196, 11.91293975274600708513615359855, 12.53044634752031087551338452622, 13.62063945302284299571071919276, 13.78590389127411452695871012487, 15.06817651846621979861295442311, 15.73103311300125968564536335025, 16.26606375024573452391811944782, 16.990634203659084921406615483259, 18.058508464461604621758773489864, 18.629458596441270719540943183187, 19.2453017056961013913176625667, 19.93788973380668701484555313789

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