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Properties

Label 2-1800-1800.931-c0-0-1
Degree $2$
Conductor $1800$
Sign $0.851 - 0.523i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
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  + (0.743 + 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 + 0.994i)4-s + (0.994 − 0.104i)5-s + (0.994 + 0.104i)6-s + (0.866 − 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.413 + 0.459i)11-s + (0.669 + 0.743i)12-s + (−1.20 + 1.08i)13-s + (0.978 + 0.207i)14-s + (0.743 − 0.669i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 + 0.994i)4-s + (0.994 − 0.104i)5-s + (0.994 + 0.104i)6-s + (0.866 − 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.413 + 0.459i)11-s + (0.669 + 0.743i)12-s + (−1.20 + 1.08i)13-s + (0.978 + 0.207i)14-s + (0.743 − 0.669i)15-s + (−0.978 + 0.207i)16-s + (−0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.851 - 0.523i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.851 - 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.474129754\)
\(L(\frac12)\) \(\approx\) \(2.474129754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.743 - 0.669i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.994 + 0.104i)T \)
good7 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.413 - 0.459i)T + (-0.104 - 0.994i)T^{2} \)
13 \( 1 + (1.20 - 1.08i)T + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.207 - 0.978i)T + (-0.913 - 0.406i)T^{2} \)
29 \( 1 + (0.251 - 0.564i)T + (-0.669 - 0.743i)T^{2} \)
31 \( 1 + (0.406 + 0.913i)T + (-0.669 + 0.743i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2} \)
53 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.104 + 0.994i)T^{2} \)
61 \( 1 + (-0.743 - 0.669i)T + (0.104 + 0.994i)T^{2} \)
67 \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.251 + 0.564i)T + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (0.0646 - 0.614i)T + (-0.978 - 0.207i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.669 + 0.743i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.166598279553728271503218639841, −8.761160904402304117621828032440, −7.65281266591967465917900704997, −7.12204618024458327440316944346, −6.61379621283506029710969383790, −5.38258117509407176660646823546, −4.71532494135022391380194252583, −3.85387529982627194594816727092, −2.34862008093504993769610103878, −2.03877755983766968420454142896, 1.88987958551792556223339476912, 2.42480169018687278073725286970, 3.30908475819232154744384149184, 4.55770667521927088884364107667, 5.09939003724839557032451518265, 5.83392625493794589668403428405, 6.84781334059969072738753144786, 8.227844445742367188647444156194, 8.575349034277811693470364086324, 9.733808949505023707766585217703

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