Properties

Label 2-1200-1.1-c3-0-3
Degree $2$
Conductor $1200$
Sign $1$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 4·7-s + 9·9-s − 72·11-s + 6·13-s − 38·17-s − 52·19-s − 12·21-s + 152·23-s − 27·27-s − 78·29-s − 120·31-s + 216·33-s + 150·37-s − 18·39-s + 362·41-s − 484·43-s + 280·47-s − 327·49-s + 114·51-s + 670·53-s + 156·57-s − 696·59-s + 222·61-s + 36·63-s − 4·67-s − 456·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.215·7-s + 1/3·9-s − 1.97·11-s + 0.128·13-s − 0.542·17-s − 0.627·19-s − 0.124·21-s + 1.37·23-s − 0.192·27-s − 0.499·29-s − 0.695·31-s + 1.13·33-s + 0.666·37-s − 0.0739·39-s + 1.37·41-s − 1.71·43-s + 0.868·47-s − 0.953·49-s + 0.313·51-s + 1.73·53-s + 0.362·57-s − 1.53·59-s + 0.465·61-s + 0.0719·63-s − 0.00729·67-s − 0.795·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.074026174\)
\(L(\frac12)\) \(\approx\) \(1.074026174\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
5 \( 1 \)
good7 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 + 72 T + p^{3} T^{2} \)
13 \( 1 - 6 T + p^{3} T^{2} \)
17 \( 1 + 38 T + p^{3} T^{2} \)
19 \( 1 + 52 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 + 78 T + p^{3} T^{2} \)
31 \( 1 + 120 T + p^{3} T^{2} \)
37 \( 1 - 150 T + p^{3} T^{2} \)
41 \( 1 - 362 T + p^{3} T^{2} \)
43 \( 1 + 484 T + p^{3} T^{2} \)
47 \( 1 - 280 T + p^{3} T^{2} \)
53 \( 1 - 670 T + p^{3} T^{2} \)
59 \( 1 + 696 T + p^{3} T^{2} \)
61 \( 1 - 222 T + p^{3} T^{2} \)
67 \( 1 + 4 T + p^{3} T^{2} \)
71 \( 1 + 96 T + p^{3} T^{2} \)
73 \( 1 + 178 T + p^{3} T^{2} \)
79 \( 1 - 8 p T + p^{3} T^{2} \)
83 \( 1 + 612 T + p^{3} T^{2} \)
89 \( 1 - 994 T + p^{3} T^{2} \)
97 \( 1 + 1634 T + p^{3} T^{2} \)
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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

−9.413546668410605712924156276610, −8.497310512738676789586005233622, −7.67479124765257075715448512391, −6.94819411649591222504256383414, −5.86472365838579624169533227947, −5.16711049966913734828213781660, −4.40244113680151271480742637498, −3.04375495940361895483561746749, −2.03237067325083737730616570480, −0.52752301244668289932876258540, 0.52752301244668289932876258540, 2.03237067325083737730616570480, 3.04375495940361895483561746749, 4.40244113680151271480742637498, 5.16711049966913734828213781660, 5.86472365838579624169533227947, 6.94819411649591222504256383414, 7.67479124765257075715448512391, 8.497310512738676789586005233622, 9.413546668410605712924156276610

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