Properties

Label 2-9600-1.1-c1-0-44
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

\(L(1)\) \(\approx\) \(2.627402493\)
\(L(\frac12)\) \(\approx\) \(2.627402493\)
\(L(\frac{3}{2})\) 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

−7.66787181904835254536280404261, −7.05288897178836521190481390606, −6.44093668351668166909188405670, −5.65872496122396304668826661032, −4.94953394822865351226801029376, −3.88801211333409031435937142165, −3.52735596718546638915648635756, −2.71294597011535288008768454291, −1.74809157682506433763351423678, −0.76911207638132933654706648277, 0.76911207638132933654706648277, 1.74809157682506433763351423678, 2.71294597011535288008768454291, 3.52735596718546638915648635756, 3.88801211333409031435937142165, 4.94953394822865351226801029376, 5.65872496122396304668826661032, 6.44093668351668166909188405670, 7.05288897178836521190481390606, 7.66787181904835254536280404261

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