Properties

Degree $2$
Conductor $9408$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.46·5-s + 9-s − 5.46·11-s + 2·13-s − 3.46·15-s − 7.46·17-s − 6.92·19-s + 5.46·23-s + 6.99·25-s + 27-s − 8.92·29-s − 2.92·31-s − 5.46·33-s + 2·37-s + 2·39-s − 4.53·41-s + 8·43-s − 3.46·45-s + 2.92·47-s − 7.46·51-s + 2·53-s + 18.9·55-s − 6.92·57-s − 14.9·59-s + 4.92·61-s − 6.92·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.54·5-s + 0.333·9-s − 1.64·11-s + 0.554·13-s − 0.894·15-s − 1.81·17-s − 1.58·19-s + 1.13·23-s + 1.39·25-s + 0.192·27-s − 1.65·29-s − 0.525·31-s − 0.951·33-s + 0.328·37-s + 0.320·39-s − 0.708·41-s + 1.21·43-s − 0.516·45-s + 0.427·47-s − 1.04·51-s + 0.274·53-s + 2.55·55-s − 0.917·57-s − 1.94·59-s + 0.630·61-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: \(0\)
Selberg data: \((2,\ 9408,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6028757861\)
\(L(\frac12)\) \(\approx\) \(0.6028757861\)
\(L(\frac{3}{2})\) 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 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
19 \( 1 + 6.92T + 19T^{2} \)
23 \( 1 - 5.46T + 23T^{2} \)
29 \( 1 + 8.92T + 29T^{2} \)
31 \( 1 + 2.92T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 2.92T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−7.65651325732566230463417525634, −7.29287451208356264095984422391, −6.53337898913859922368373281997, −5.59189357643917127795823790665, −4.61628337285320741375164272825, −4.25312523685626085477505613103, −3.46540363544261841324593555692, −2.68514697708499295060152238655, −1.93407079451544467341685431602, −0.33798999570611027605643578164, 0.33798999570611027605643578164, 1.93407079451544467341685431602, 2.68514697708499295060152238655, 3.46540363544261841324593555692, 4.25312523685626085477505613103, 4.61628337285320741375164272825, 5.59189357643917127795823790665, 6.53337898913859922368373281997, 7.29287451208356264095984422391, 7.65651325732566230463417525634

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