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Properties

Label 2-1800-200.197-c0-0-0
Degree $2$
Conductor $1800$
Sign $-0.860 - 0.509i$
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  + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.891 + 0.453i)5-s + (1.39 + 1.39i)7-s + (−0.987 − 0.156i)8-s + (−0.809 − 0.587i)10-s + (0.863 + 0.280i)11-s + (−0.610 + 1.87i)14-s + (−0.309 − 0.951i)16-s + (0.156 − 0.987i)20-s + (0.142 + 0.896i)22-s + (0.587 − 0.809i)25-s + (−1.95 + 0.309i)28-s + (−1.14 − 0.831i)29-s + (−0.951 + 0.690i)31-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (−0.587 + 0.809i)4-s + (−0.891 + 0.453i)5-s + (1.39 + 1.39i)7-s + (−0.987 − 0.156i)8-s + (−0.809 − 0.587i)10-s + (0.863 + 0.280i)11-s + (−0.610 + 1.87i)14-s + (−0.309 − 0.951i)16-s + (0.156 − 0.987i)20-s + (0.142 + 0.896i)22-s + (0.587 − 0.809i)25-s + (−1.95 + 0.309i)28-s + (−1.14 − 0.831i)29-s + (−0.951 + 0.690i)31-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.860 - 0.509i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.860 - 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288321654\)
\(L(\frac12)\) \(\approx\) \(1.288321654\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 - 0.891i)T \)
3 \( 1 \)
5 \( 1 + (0.891 - 0.453i)T \)
good7 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
11 \( 1 + (-0.863 - 0.280i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 + 0.809i)T^{2} \)
29 \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.297 - 1.87i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.642 + 1.26i)T + (-0.587 + 0.809i)T^{2} \)
79 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (-1.59 - 0.253i)T + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \)
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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

−9.246883889261492964777244550693, −8.928078395260447889245947628349, −7.929432490396753994560279727263, −7.62021481145295059361336799458, −6.57479218654022018702834855886, −5.77776435287569298791109558144, −4.93280836760983396324035117773, −4.21888393545930830066480682473, −3.23704445615472740035383897178, −2.00161302584921422221438921037, 0.920013236065428239616935050008, 1.82901514207617851340193793968, 3.58054275344553991565452221319, 3.99204477266767637439903884486, 4.76282133606476899253619575552, 5.52382713231716710333909135926, 6.89048399857473243698078830954, 7.62356503577734419859228900525, 8.458947768421540257855065044176, 9.139571232646138537023690261777

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