Properties

Label 2-9576-1.1-c1-0-67
Degree $2$
Conductor $9576$
Sign $-1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·5-s − 7-s − 2·13-s + 3.46·17-s − 19-s − 4·23-s + 6.99·25-s + 0.535·29-s + 10.9·31-s + 3.46·35-s − 2·37-s − 6·41-s + 6.92·43-s − 9.46·47-s + 49-s + 3.46·53-s − 4·59-s + 6·61-s + 6.92·65-s + 14.9·67-s − 13.4·71-s − 0.928·73-s + 5.46·83-s − 11.9·85-s + 15.8·89-s + 2·91-s + 3.46·95-s + ⋯
L(s)  = 1  − 1.54·5-s − 0.377·7-s − 0.554·13-s + 0.840·17-s − 0.229·19-s − 0.834·23-s + 1.39·25-s + 0.0995·29-s + 1.96·31-s + 0.585·35-s − 0.328·37-s − 0.937·41-s + 1.05·43-s − 1.38·47-s + 0.142·49-s + 0.475·53-s − 0.520·59-s + 0.768·61-s + 0.859·65-s + 1.82·67-s − 1.59·71-s − 0.108·73-s + 0.599·83-s − 1.30·85-s + 1.68·89-s + 0.209·91-s + 0.355·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9576\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-1$
Analytic conductor: \(76.4647\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
19 \( 1 + T \)
good5 \( 1 + 3.46T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 0.535T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 14.9T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 0.928T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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

−7.49319518637628585033979555324, −6.73317494989573537807050785996, −6.12786390766243469516063658362, −5.10645268147226859492584833635, −4.51764354524026810845626081871, −3.75900609362930158354664743603, −3.21796959530136593973094686599, −2.31929969940564237489543718826, −0.967189917001944612391575457290, 0, 0.967189917001944612391575457290, 2.31929969940564237489543718826, 3.21796959530136593973094686599, 3.75900609362930158354664743603, 4.51764354524026810845626081871, 5.10645268147226859492584833635, 6.12786390766243469516063658362, 6.73317494989573537807050785996, 7.49319518637628585033979555324

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