Properties

Label 2-1800-1.1-c1-0-21
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  + 4·11-s − 6·13-s − 6·17-s − 4·19-s + 2·29-s − 8·31-s + 2·37-s + 6·41-s − 12·43-s + 8·47-s − 7·49-s + 6·53-s − 12·59-s + 14·61-s − 4·67-s − 8·71-s + 6·73-s − 8·79-s − 12·83-s − 10·89-s − 2·97-s − 6·101-s − 4·107-s − 18·109-s − 6·113-s + ⋯
L(s)  = 1  + 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.917·19-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s + 0.824·53-s − 1.56·59-s + 1.79·61-s − 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.900·79-s − 1.31·83-s − 1.05·89-s − 0.203·97-s − 0.597·101-s − 0.386·107-s − 1.72·109-s − 0.564·113-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.006677185559548999172787233240, −8.167325496630191558293323555990, −7.06475469847576410250018540799, −6.71920497811244846696035723021, −5.65229906256087841325445637569, −4.60954800977677454684167974744, −4.03205568616657235432062033061, −2.69982664061727220731686131413, −1.76966313536701547243535553599, 0, 1.76966313536701547243535553599, 2.69982664061727220731686131413, 4.03205568616657235432062033061, 4.60954800977677454684167974744, 5.65229906256087841325445637569, 6.71920497811244846696035723021, 7.06475469847576410250018540799, 8.167325496630191558293323555990, 9.006677185559548999172787233240

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