Properties

Label 2-9600-1.1-c1-0-18
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 − 6·11-s − 2·13-s + 2·17-s − 2·19-s − 2·21-s + 6·23-s + 27-s − 10·29-s + 4·31-s − 6·33-s + 2·37-s − 2·39-s + 6·41-s + 6·47-s − 3·49-s + 2·51-s − 10·53-s − 2·57-s + 6·59-s − 6·61-s − 2·63-s + 16·67-s + 6·69-s − 8·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.485·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.85·29-s + 0.718·31-s − 1.04·33-s + 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.875·47-s − 3/7·49-s + 0.280·51-s − 1.37·53-s − 0.264·57-s + 0.781·59-s − 0.768·61-s − 0.251·63-s + 1.95·67-s + 0.722·69-s − 0.949·71-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.522551255\)
\(L(\frac12)\) \(\approx\) \(1.522551255\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.63673307457394675109969495567, −7.24363430315120775271601585969, −6.35394015861602454189251739401, −5.56052832606525336238042110966, −4.98943909657996553372240376787, −4.14780051953416484392191123229, −3.17542851103246843339167056831, −2.76260796484964071492101944422, −1.94880012443427660192473848218, −0.54105666062457426769681416030, 0.54105666062457426769681416030, 1.94880012443427660192473848218, 2.76260796484964071492101944422, 3.17542851103246843339167056831, 4.14780051953416484392191123229, 4.98943909657996553372240376787, 5.56052832606525336238042110966, 6.35394015861602454189251739401, 7.24363430315120775271601585969, 7.63673307457394675109969495567

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