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Properties

Label 2-1800-1800.931-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5887126009\)
\(L(\frac12)\) \(\approx\) \(0.5887126009\)
\(L(1)\) 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

−8.863424082088983682809144353771, −8.506232029666333233916883038918, −7.889704196978169466386638721286, −6.86237847842204951730334455405, −6.49994868894441304259862104899, −4.73264488966425269157801326105, −3.61754481188853586988950060135, −3.03454148459537247483207469615, −2.19354483635704203801513653927, −0.50985980137464453530380470028, 1.61922432505441860853942633956, 3.17286063528004763782203761606, 4.00361320507803825237374295870, 4.72210176711896253924244599698, 6.10057884301191050481513746300, 6.73851104267984359919627495643, 7.65233953762706411129931400970, 8.374778000786076228579629624974, 8.785136106852429537099883424854, 9.589962371849454431827370334056

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