Properties

Label 2-1800-1.1-c1-0-16
Degree $2$
Conductor $1800$
Sign $-1$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s + 2·11-s − 4·13-s + 2·17-s + 4·19-s − 8·23-s − 10·29-s + 4·31-s + 8·43-s − 8·47-s − 3·49-s − 6·53-s − 14·59-s − 14·61-s + 4·67-s + 12·71-s − 6·73-s − 4·77-s − 12·79-s − 4·83-s − 12·89-s + 8·91-s + 14·97-s + 6·101-s + 14·103-s + 12·107-s + 2·109-s + ⋯
L(s)  = 1  − 0.755·7-s + 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 1.85·29-s + 0.718·31-s + 1.21·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s − 1.82·59-s − 1.79·61-s + 0.488·67-s + 1.42·71-s − 0.702·73-s − 0.455·77-s − 1.35·79-s − 0.439·83-s − 1.27·89-s + 0.838·91-s + 1.42·97-s + 0.597·101-s + 1.37·103-s + 1.16·107-s + 0.191·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.145483992614508398241889104525, −7.86913692024763332060144721483, −7.45065071242999963428388622511, −6.39891699284620141219634951096, −5.77511827593267937653674507024, −4.74288937907914556693961913412, −3.76449264160662743485897571369, −2.90937812661793913025246512240, −1.67682860274284527232402663231, 0, 1.67682860274284527232402663231, 2.90937812661793913025246512240, 3.76449264160662743485897571369, 4.74288937907914556693961913412, 5.77511827593267937653674507024, 6.39891699284620141219634951096, 7.45065071242999963428388622511, 7.86913692024763332060144721483, 9.145483992614508398241889104525

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