Properties

Label 2-1800-1.1-c3-0-65
Degree $2$
Conductor $1800$
Sign $-1$
Analytic cond. $106.203$
Root an. cond. $10.3055$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·7-s + 16·11-s + 6·13-s − 6·17-s − 124·19-s + 42·23-s − 142·29-s − 188·31-s − 202·37-s − 54·41-s − 66·43-s + 38·47-s − 19·49-s + 738·53-s − 564·59-s − 262·61-s + 554·67-s − 140·71-s − 882·73-s + 288·77-s − 1.16e3·79-s + 642·83-s + 854·89-s + 108·91-s + 478·97-s + 1.79e3·101-s − 642·103-s + ⋯
L(s)  = 1  + 0.971·7-s + 0.438·11-s + 0.128·13-s − 0.0856·17-s − 1.49·19-s + 0.380·23-s − 0.909·29-s − 1.08·31-s − 0.897·37-s − 0.205·41-s − 0.234·43-s + 0.117·47-s − 0.0553·49-s + 1.91·53-s − 1.24·59-s − 0.549·61-s + 1.01·67-s − 0.234·71-s − 1.41·73-s + 0.426·77-s − 1.65·79-s + 0.849·83-s + 1.01·89-s + 0.124·91-s + 0.500·97-s + 1.76·101-s − 0.614·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(106.203\)
Root analytic conductor: \(10.3055\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 18 T + p^{3} T^{2} \)
11 \( 1 - 16 T + p^{3} T^{2} \)
13 \( 1 - 6 T + p^{3} T^{2} \)
17 \( 1 + 6 T + p^{3} T^{2} \)
19 \( 1 + 124 T + p^{3} T^{2} \)
23 \( 1 - 42 T + p^{3} T^{2} \)
29 \( 1 + 142 T + p^{3} T^{2} \)
31 \( 1 + 188 T + p^{3} T^{2} \)
37 \( 1 + 202 T + p^{3} T^{2} \)
41 \( 1 + 54 T + p^{3} T^{2} \)
43 \( 1 + 66 T + p^{3} T^{2} \)
47 \( 1 - 38 T + p^{3} T^{2} \)
53 \( 1 - 738 T + p^{3} T^{2} \)
59 \( 1 + 564 T + p^{3} T^{2} \)
61 \( 1 + 262 T + p^{3} T^{2} \)
67 \( 1 - 554 T + p^{3} T^{2} \)
71 \( 1 + 140 T + p^{3} T^{2} \)
73 \( 1 + 882 T + p^{3} T^{2} \)
79 \( 1 + 1160 T + p^{3} T^{2} \)
83 \( 1 - 642 T + p^{3} T^{2} \)
89 \( 1 - 854 T + p^{3} T^{2} \)
97 \( 1 - 478 T + p^{3} T^{2} \)
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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

−8.677032330280436954302927106022, −7.73151981954266855813602480501, −7.00437159362150361684871153334, −6.09501456817915574557965042986, −5.22087021827547910939606776794, −4.37983659672026445232459538768, −3.56884531776830265114822156562, −2.21921290175955667938471934862, −1.43657579461213153138551171854, 0, 1.43657579461213153138551171854, 2.21921290175955667938471934862, 3.56884531776830265114822156562, 4.37983659672026445232459538768, 5.22087021827547910939606776794, 6.09501456817915574557965042986, 7.00437159362150361684871153334, 7.73151981954266855813602480501, 8.677032330280436954302927106022

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