Properties

Label 2-9576-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.23·5-s − 7-s + 5.23·11-s − 0.763·13-s − 7.23·17-s + 19-s + 3.23·23-s − 3.47·25-s − 8.47·29-s + 0.472·31-s − 1.23·35-s + 8.94·37-s − 2·41-s − 4·43-s + 6.47·47-s + 49-s + 0.472·53-s + 6.47·55-s + 8·59-s + 8.47·61-s − 0.944·65-s + 0.763·67-s + 1.52·71-s + 4.47·73-s − 5.23·77-s + 15.7·79-s − 2·83-s + ⋯
L(s)  = 1  + 0.552·5-s − 0.377·7-s + 1.57·11-s − 0.211·13-s − 1.75·17-s + 0.229·19-s + 0.674·23-s − 0.694·25-s − 1.57·29-s + 0.0847·31-s − 0.208·35-s + 1.47·37-s − 0.312·41-s − 0.609·43-s + 0.944·47-s + 0.142·49-s + 0.0648·53-s + 0.872·55-s + 1.04·59-s + 1.08·61-s − 0.117·65-s + 0.0933·67-s + 0.181·71-s + 0.523·73-s − 0.596·77-s + 1.76·79-s − 0.219·83-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: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.194295092\)
\(L(\frac12)\) \(\approx\) \(2.194295092\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 + 0.763T + 13T^{2} \)
17 \( 1 + 7.23T + 17T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 + 8.47T + 29T^{2} \)
31 \( 1 - 0.472T + 31T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 - 0.763T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 1.23T + 97T^{2} \)
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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

−7.55695829701518011535743341331, −6.84158340841337677529916553504, −6.42050509685313391210844701869, −5.76824538076009081581471509434, −4.92582765674797315601368394393, −4.08577905614069686340656597051, −3.59705079604356304886173667533, −2.42948670189715872435204141802, −1.84007508218933729871461112504, −0.70503064142347803198283574142, 0.70503064142347803198283574142, 1.84007508218933729871461112504, 2.42948670189715872435204141802, 3.59705079604356304886173667533, 4.08577905614069686340656597051, 4.92582765674797315601368394393, 5.76824538076009081581471509434, 6.42050509685313391210844701869, 6.84158340841337677529916553504, 7.55695829701518011535743341331

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