Properties

Degree $2$
Conductor $9408$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 0.454·5-s + 9-s + 5.79·11-s − 5.88·13-s + 0.454·15-s − 2.90·17-s + 5.88·19-s − 2.90·23-s − 4.79·25-s + 27-s − 3.54·29-s + 4.33·31-s + 5.79·33-s − 7.70·37-s − 5.88·39-s − 9.58·41-s − 10.7·43-s + 0.454·45-s − 4.90·47-s − 2.90·51-s − 13.1·53-s + 2.63·55-s + 5.88·57-s − 1.79·59-s + 4.67·61-s − 2.67·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.203·5-s + 0.333·9-s + 1.74·11-s − 1.63·13-s + 0.117·15-s − 0.705·17-s + 1.34·19-s − 0.606·23-s − 0.958·25-s + 0.192·27-s − 0.658·29-s + 0.779·31-s + 1.00·33-s − 1.26·37-s − 0.942·39-s − 1.49·41-s − 1.64·43-s + 0.0678·45-s − 0.716·47-s − 0.407·51-s − 1.80·53-s + 0.355·55-s + 0.779·57-s − 0.233·59-s + 0.598·61-s − 0.331·65-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{9408} (1, \cdot )$
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 \( 1 - 0.454T + 5T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + 2.90T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + 2.90T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 - 4.33T + 31T^{2} \)
37 \( 1 + 7.70T + 37T^{2} \)
41 \( 1 + 9.58T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + 4.90T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 1.79T + 59T^{2} \)
61 \( 1 - 4.67T + 61T^{2} \)
67 \( 1 + 7.88T + 67T^{2} \)
71 \( 1 + 0.909T + 71T^{2} \)
73 \( 1 + 5.20T + 73T^{2} \)
79 \( 1 + 2.75T + 79T^{2} \)
83 \( 1 - 9.97T + 83T^{2} \)
89 \( 1 - 4.90T + 89T^{2} \)
97 \( 1 - 5.79T + 97T^{2} \)
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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

−7.34020726946861713556109191396, −6.75587048098838539261431354111, −6.17267145425137821830976439875, −5.11233823799644057809140436418, −4.62870180867377877754050293402, −3.67229119775004811894662687036, −3.16635104865747422298085471866, −2.02668273010612640987199235782, −1.52775281735858768905543728517, 0, 1.52775281735858768905543728517, 2.02668273010612640987199235782, 3.16635104865747422298085471866, 3.67229119775004811894662687036, 4.62870180867377877754050293402, 5.11233823799644057809140436418, 6.17267145425137821830976439875, 6.75587048098838539261431354111, 7.34020726946861713556109191396

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