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 + 3.46·5-s + 9-s − 1.46·11-s + 2·13-s − 3.46·15-s − 0.535·17-s − 6.92·19-s + 1.46·23-s + 6.99·25-s − 27-s + 4.92·29-s − 10.9·31-s + 1.46·33-s + 2·37-s − 2·39-s − 11.4·41-s − 8·43-s + 3.46·45-s + 10.9·47-s + 0.535·51-s + 2·53-s − 5.07·55-s + 6.92·57-s + 1.07·59-s − 8.92·61-s + 6.92·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.54·5-s + 0.333·9-s − 0.441·11-s + 0.554·13-s − 0.894·15-s − 0.129·17-s − 1.58·19-s + 0.305·23-s + 1.39·25-s − 0.192·27-s + 0.915·29-s − 1.96·31-s + 0.254·33-s + 0.328·37-s − 0.320·39-s − 1.79·41-s − 1.21·43-s + 0.516·45-s + 1.59·47-s + 0.0750·51-s + 0.274·53-s − 0.683·55-s + 0.917·57-s + 0.139·59-s − 1.14·61-s + 0.859·65-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 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 0.535T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 - 4.92T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + 2.92T + 67T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−7.09536912984812111731608418318, −6.54452745151342578597740221740, −5.98047010335366169344027546632, −5.42394772179762228906445326732, −4.79273265910832254025447315884, −3.92007417955532563623689185237, −2.87282502434015661290993280669, −2.01741251832874320211315377965, −1.40951223292203075536820948492, 0, 1.40951223292203075536820948492, 2.01741251832874320211315377965, 2.87282502434015661290993280669, 3.92007417955532563623689185237, 4.79273265910832254025447315884, 5.42394772179762228906445326732, 5.98047010335366169344027546632, 6.54452745151342578597740221740, 7.09536912984812111731608418318

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