Properties

Label 2-9600-1.1-c1-0-120
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 4·7-s + 9-s + 6·13-s − 2·17-s + 6·19-s − 4·21-s − 6·23-s + 27-s − 8·29-s − 8·31-s + 10·37-s + 6·39-s − 6·41-s + 4·43-s − 2·47-s + 9·49-s − 2·51-s − 6·53-s + 6·57-s − 12·59-s + 14·61-s − 4·63-s + 4·67-s − 6·69-s − 8·71-s − 4·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s + 1.37·19-s − 0.872·21-s − 1.25·23-s + 0.192·27-s − 1.48·29-s − 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 0.794·57-s − 1.56·59-s + 1.79·61-s − 0.503·63-s + 0.488·67-s − 0.722·69-s − 0.949·71-s − 0.468·73-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: \(1\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−7.42821189229031598906405889344, −6.61324747593383215936433165243, −6.02830278465784358925870839173, −5.53307504715960159764472147716, −4.28486347853939884771274031693, −3.56029124104524321077881421582, −3.30634596566881086087887394666, −2.24676611100900259902487612788, −1.28020943603786707112377428408, 0, 1.28020943603786707112377428408, 2.24676611100900259902487612788, 3.30634596566881086087887394666, 3.56029124104524321077881421582, 4.28486347853939884771274031693, 5.53307504715960159764472147716, 6.02830278465784358925870839173, 6.61324747593383215936433165243, 7.42821189229031598906405889344

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