Properties

Label 2-9600-1.1-c1-0-7
Degree $2$
Conductor $9600$
Sign $1$
Analytic cond. $76.6563$
Root an. cond. $8.75536$
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 − 2·7-s + 9-s − 4·11-s + 2·13-s + 2·17-s − 8·19-s + 2·21-s − 4·23-s − 27-s − 6·31-s + 4·33-s − 2·37-s − 2·39-s + 6·41-s − 4·47-s − 3·49-s − 2·51-s + 8·57-s + 4·59-s − 14·61-s − 2·63-s + 4·67-s + 4·69-s + 12·71-s + 10·73-s + 8·77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.07·31-s + 0.696·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.05·57-s + 0.520·59-s − 1.79·61-s − 0.251·63-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + 1.17·73-s + 0.911·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9600\)    =    \(2^{7} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(76.6563\)
Root analytic conductor: \(8.75536\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{9600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6192847067\)
\(L(\frac12)\) \(\approx\) \(0.6192847067\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.73815604164369169821342098956, −6.85101840841362537104427690689, −6.27313190421731068099902685814, −5.73364318390344893732026132208, −5.03707169329726589736408769710, −4.17603012762576770211668628573, −3.51969663080255673561059359971, −2.57264458953716636539081361760, −1.74725810428037864412471774362, −0.37131919747375929898043950705, 0.37131919747375929898043950705, 1.74725810428037864412471774362, 2.57264458953716636539081361760, 3.51969663080255673561059359971, 4.17603012762576770211668628573, 5.03707169329726589736408769710, 5.73364318390344893732026132208, 6.27313190421731068099902685814, 6.85101840841362537104427690689, 7.73815604164369169821342098956

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