Properties

Label 2-1800-1.1-c1-0-9
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s − 4·11-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 2·29-s − 8·31-s − 6·37-s + 6·41-s + 8·43-s + 4·47-s + 9·49-s + 6·53-s + 4·59-s − 2·61-s − 8·67-s + 6·73-s − 16·77-s − 16·83-s + 6·89-s + 8·91-s + 14·97-s − 6·101-s − 4·103-s + 14·109-s + 18·113-s + ⋯
L(s)  = 1  + 1.51·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.520·59-s − 0.256·61-s − 0.977·67-s + 0.702·73-s − 1.82·77-s − 1.75·83-s + 0.635·89-s + 0.838·91-s + 1.42·97-s − 0.597·101-s − 0.394·103-s + 1.34·109-s + 1.69·113-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: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.085116964\)
\(L(\frac12)\) \(\approx\) \(2.085116964\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−9.100428369436491434363376978977, −8.469028469012419436497301616644, −7.61379691617992725579918349351, −7.25922965439850703725280828805, −5.74675556302869039795734872495, −5.29208484858772092309966421313, −4.45286390316500807643919809296, −3.31004324327388789633257437687, −2.19847701489412229516994368525, −1.05539838708270688643647945003, 1.05539838708270688643647945003, 2.19847701489412229516994368525, 3.31004324327388789633257437687, 4.45286390316500807643919809296, 5.29208484858772092309966421313, 5.74675556302869039795734872495, 7.25922965439850703725280828805, 7.61379691617992725579918349351, 8.469028469012419436497301616644, 9.100428369436491434363376978977

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