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Properties

Label 2-1800-360.259-c0-0-3
Degree $2$
Conductor $1800$
Sign $0.993 + 0.114i$
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  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999i·12-s + (−0.5 + 0.866i)16-s − i·17-s − 0.999i·18-s + 19-s + (−0.866 + 0.499i)22-s + (0.5 − 0.866i)24-s + 0.999i·27-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + 0.999i·12-s + (−0.5 + 0.866i)16-s − i·17-s − 0.999i·18-s + 19-s + (−0.866 + 0.499i)22-s + (0.5 − 0.866i)24-s + 0.999i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096487611\)
\(L(\frac12)\) \(\approx\) \(1.096487611\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.401547013888780752027641485408, −8.862388066119606818922258422997, −8.066169273326099602441261665573, −7.43719146216474774232026431621, −6.54949001973275204253985328822, −5.28159038895397065606225828120, −4.15073783244260076117080371187, −3.27781288724194411019517302644, −2.62082327897373365573977006449, −1.27273913455544841842104980589, 1.33213616736614162269445845775, 2.19142295896213575960751981334, 3.41756445064534458570763718293, 4.56973642910548609658246940159, 5.77045663729208066589784318353, 6.60257677060489823061898846934, 7.30078349076498136544696965429, 7.87474238946064128015299790298, 8.698128939856023349939267827880, 9.300927179036082178703922463949

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