Properties

Label 2-9408-1.1-c1-0-108
Degree $2$
Conductor $9408$
Sign $-1$
Analytic cond. $75.1232$
Root an. cond. $8.66736$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s − 3·11-s − 4·13-s − 15-s + 4·19-s + 8·23-s − 4·25-s − 27-s + 3·29-s − 5·31-s + 3·33-s − 8·37-s + 4·39-s + 8·41-s − 6·43-s + 45-s + 10·47-s − 9·53-s − 3·55-s − 4·57-s + 5·59-s + 10·61-s − 4·65-s − 6·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.258·15-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.898·31-s + 0.522·33-s − 1.31·37-s + 0.640·39-s + 1.24·41-s − 0.914·43-s + 0.149·45-s + 1.45·47-s − 1.23·53-s − 0.404·55-s − 0.529·57-s + 0.650·59-s + 1.28·61-s − 0.496·65-s − 0.733·67-s − 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(75.1232\)
Root analytic conductor: \(8.66736\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−7.25556380379341063813322432424, −6.81155066983315445218550683284, −5.82128168525755754297443293137, −5.21798747405709523541052327426, −4.94369911770872072111273594853, −3.85805870080380911684721325744, −2.91334044941297982903056670611, −2.22871318027998081417226803477, −1.14384835958507825698585314815, 0, 1.14384835958507825698585314815, 2.22871318027998081417226803477, 2.91334044941297982903056670611, 3.85805870080380911684721325744, 4.94369911770872072111273594853, 5.21798747405709523541052327426, 5.82128168525755754297443293137, 6.81155066983315445218550683284, 7.25556380379341063813322432424

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