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 − 4·5-s + 9-s − 6·11-s + 5·13-s + 4·15-s − 2·17-s + 19-s − 6·23-s + 11·25-s − 27-s + 3·31-s + 6·33-s − 3·37-s − 5·39-s + 6·41-s − 5·43-s − 4·45-s + 4·47-s + 2·51-s + 6·53-s + 24·55-s − 57-s − 6·59-s − 2·61-s − 20·65-s − 7·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s + 1/3·9-s − 1.80·11-s + 1.38·13-s + 1.03·15-s − 0.485·17-s + 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.538·31-s + 1.04·33-s − 0.493·37-s − 0.800·39-s + 0.937·41-s − 0.762·43-s − 0.596·45-s + 0.583·47-s + 0.280·51-s + 0.824·53-s + 3.23·55-s − 0.132·57-s − 0.781·59-s − 0.256·61-s − 2.48·65-s − 0.855·67-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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.55373200800050251421950752831, −6.71519726514733804353725694783, −6.00719521772456662099092194019, −5.21276374325913581601746580852, −4.55106162331997349526435963011, −3.85983704785335399228137277587, −3.25737926538785744000407547749, −2.23862928235213562615229519758, −0.839955276776077289904040619176, 0, 0.839955276776077289904040619176, 2.23862928235213562615229519758, 3.25737926538785744000407547749, 3.85983704785335399228137277587, 4.55106162331997349526435963011, 5.21276374325913581601746580852, 6.00719521772456662099092194019, 6.71519726514733804353725694783, 7.55373200800050251421950752831

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