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Properties

Label 2-1800-200.117-c0-0-0
Degree $2$
Conductor $1800$
Sign $-0.509 - 0.860i$
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  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.453 + 0.891i)5-s + (0.221 + 0.221i)7-s + (0.156 + 0.987i)8-s + (−0.809 + 0.587i)10-s + (−1.69 + 0.550i)11-s + (0.0966 + 0.297i)14-s + (−0.309 + 0.951i)16-s + (−0.987 + 0.156i)20-s + (−1.76 − 0.278i)22-s + (−0.587 − 0.809i)25-s + (−0.0489 + 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  + (0.891 + 0.453i)2-s + (0.587 + 0.809i)4-s + (−0.453 + 0.891i)5-s + (0.221 + 0.221i)7-s + (0.156 + 0.987i)8-s + (−0.809 + 0.587i)10-s + (−1.69 + 0.550i)11-s + (0.0966 + 0.297i)14-s + (−0.309 + 0.951i)16-s + (−0.987 + 0.156i)20-s + (−1.76 − 0.278i)22-s + (−0.587 − 0.809i)25-s + (−0.0489 + 0.309i)28-s + (1.14 − 0.831i)29-s + (0.951 + 0.690i)31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.509 - 0.860i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ -0.509 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.601481698\)
\(L(\frac12)\) \(\approx\) \(1.601481698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.891 - 0.453i)T \)
3 \( 1 \)
5 \( 1 + (0.453 - 0.891i)T \)
good7 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
11 \( 1 + (1.69 - 0.550i)T + (0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.87 - 0.297i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
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 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.253 + 1.59i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \)
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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

−10.02679650656277731610897389054, −8.497741975866952876605286768286, −7.992874618951924027713158079277, −7.25718021106848805511204790466, −6.59240359705719765718795971117, −5.63379102557485639254939334052, −4.87400598322364067600371289712, −4.02352452197665808942371487129, −2.87079209459428324785436684459, −2.36566309277354228844809461402, 0.896450814689238619868777663188, 2.35017735110632316378013637308, 3.29803679383944395816264421087, 4.33718214209444322408194956760, 5.04027784150727425281698400307, 5.61822522854002364295917468998, 6.67507365259873140997217168094, 7.74412406533713419038484077592, 8.257796353842704866286069838855, 9.288442479814118333059614213658

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