Properties

Label 2-9576-1.1-c1-0-66
Degree $2$
Conductor $9576$
Sign $1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
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  + 2·5-s − 7-s + 4·11-s + 2·13-s + 6·17-s + 19-s − 25-s + 10·29-s + 8·31-s − 2·35-s − 2·37-s − 6·41-s + 4·43-s + 8·47-s + 49-s − 6·53-s + 8·55-s + 4·59-s − 10·61-s + 4·65-s + 8·67-s − 12·71-s − 14·73-s − 4·77-s − 4·79-s + 4·83-s + 12·85-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.229·19-s − 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + 0.977·67-s − 1.42·71-s − 1.63·73-s − 0.455·77-s − 0.450·79-s + 0.439·83-s + 1.30·85-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.178757548\)
\(L(\frac12)\) \(\approx\) \(3.178757548\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.65478047116696422669303448072, −6.87000862610472789045071680173, −6.16482505023945957063098815526, −5.91960408483394904514317921926, −4.96762823348528019770922126026, −4.18290305064466179578328771321, −3.34584100967352602302155182341, −2.70134925506296298553437922558, −1.53860957248944911385408617949, −0.953031729197470419184280734853, 0.953031729197470419184280734853, 1.53860957248944911385408617949, 2.70134925506296298553437922558, 3.34584100967352602302155182341, 4.18290305064466179578328771321, 4.96762823348528019770922126026, 5.91960408483394904514317921926, 6.16482505023945957063098815526, 6.87000862610472789045071680173, 7.65478047116696422669303448072

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