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Properties

Label 2-96600-1.1-c1-0-18
Degree $2$
Conductor $96600$
Sign $1$
Analytic cond. $771.354$
Root an. cond. $27.7732$
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  + 3-s − 7-s + 9-s + 6·13-s − 2·17-s − 4·19-s − 21-s − 23-s + 27-s − 2·29-s + 4·31-s − 2·37-s + 6·39-s − 6·41-s − 4·43-s + 8·47-s + 49-s − 2·51-s − 10·53-s − 4·57-s − 2·61-s − 63-s − 4·67-s − 69-s − 6·73-s + 4·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.328·37-s + 0.960·39-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s − 0.256·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 0.702·73-s + 0.450·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(96600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(771.354\)
Root analytic conductor: \(27.7732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 96600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488818372\)
\(L(\frac12)\) \(\approx\) \(2.488818372\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.66679893153073, −13.48107278553662, −12.81921795586799, −12.54169505574822, −11.76270060568441, −11.35849803712042, −10.70010404534306, −10.43892489976812, −9.813398916916716, −9.200537792256171, −8.748660791044845, −8.407712957395867, −7.913659153700264, −7.211510626088241, −6.628379486725272, −6.240752998213015, −5.752338344202760, −4.955238835833442, −4.285321358982552, −3.873740452380174, −3.262173726242548, −2.744736447190173, −1.893076711273969, −1.444537642342988, −0.4728042113617345, 0.4728042113617345, 1.444537642342988, 1.893076711273969, 2.744736447190173, 3.262173726242548, 3.873740452380174, 4.285321358982552, 4.955238835833442, 5.752338344202760, 6.240752998213015, 6.628379486725272, 7.211510626088241, 7.913659153700264, 8.407712957395867, 8.748660791044845, 9.200537792256171, 9.813398916916716, 10.43892489976812, 10.70010404534306, 11.35849803712042, 11.76270060568441, 12.54169505574822, 12.81921795586799, 13.48107278553662, 13.66679893153073

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