Properties

Label 2-9600-1.1-c1-0-12
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 − 6·13-s − 6·17-s − 2·21-s − 4·23-s + 27-s + 4·29-s + 10·31-s − 4·33-s − 2·37-s − 6·39-s − 2·41-s − 8·43-s + 12·47-s − 3·49-s − 6·51-s + 12·53-s − 4·59-s + 2·61-s − 2·63-s − 4·67-s − 4·69-s − 4·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s − 0.436·21-s − 0.834·23-s + 0.192·27-s + 0.742·29-s + 1.79·31-s − 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s + 1.75·47-s − 3/7·49-s − 0.840·51-s + 1.64·53-s − 0.520·59-s + 0.256·61-s − 0.251·63-s − 0.488·67-s − 0.481·69-s − 0.474·71-s + 1.17·73-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: $\chi_{9600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.131820915\)
\(L(\frac12)\) \(\approx\) \(1.131820915\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.67836028948554475001548126371, −7.03797216527515100553413964284, −6.50157649314422002981040156045, −5.61696686066097488250891132837, −4.74878140873420572672293755763, −4.34325134938992504250556000135, −3.20859138627074280559035598943, −2.56433778811276467087411373879, −2.10849281981293727260394437440, −0.45397119587697427734426750720, 0.45397119587697427734426750720, 2.10849281981293727260394437440, 2.56433778811276467087411373879, 3.20859138627074280559035598943, 4.34325134938992504250556000135, 4.74878140873420572672293755763, 5.61696686066097488250891132837, 6.50157649314422002981040156045, 7.03797216527515100553413964284, 7.67836028948554475001548126371

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