Properties

Label 2-9600-1.1-c1-0-23
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

Downloads

Learn more about

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

\(L(1)\) \(\approx\) \(1.517055573\)
\(L(\frac12)\) \(\approx\) \(1.517055573\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.53668207412011919659530243812, −6.84445230805606459667372944565, −6.58739690122532465366966434700, −5.45362848812274335026775129730, −4.94477611665424568435490277562, −4.38194054884316272425505911313, −3.58531873675065089773666376742, −2.38347834548705836981541812186, −1.75846579157964817591495016051, −0.60361417218253467851429519685, 0.60361417218253467851429519685, 1.75846579157964817591495016051, 2.38347834548705836981541812186, 3.58531873675065089773666376742, 4.38194054884316272425505911313, 4.94477611665424568435490277562, 5.45362848812274335026775129730, 6.58739690122532465366966434700, 6.84445230805606459667372944565, 7.53668207412011919659530243812

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