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 + 2·5-s + 9-s − 2·11-s − 13-s + 2·15-s + 19-s − 25-s + 27-s − 4·29-s − 9·31-s − 2·33-s − 3·37-s − 39-s − 10·41-s + 5·43-s + 2·45-s + 6·47-s − 12·53-s − 4·55-s + 57-s − 12·59-s − 10·61-s − 2·65-s − 5·67-s + 6·71-s − 3·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.516·15-s + 0.229·19-s − 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.61·31-s − 0.348·33-s − 0.493·37-s − 0.160·39-s − 1.56·41-s + 0.762·43-s + 0.298·45-s + 0.875·47-s − 1.64·53-s − 0.539·55-s + 0.132·57-s − 1.56·59-s − 1.28·61-s − 0.248·65-s − 0.610·67-s + 0.712·71-s − 0.351·73-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
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

−7.55071437538943836623405220284, −6.70438184457472864541167980056, −5.93360966412049476443514792853, −5.34295079124919823511844933757, −4.66125794347334975754976010290, −3.66643801060135496247185274273, −3.00604846405785718442436393902, −2.09629495260824139671739496187, −1.56612496985569481548152977577, 0, 1.56612496985569481548152977577, 2.09629495260824139671739496187, 3.00604846405785718442436393902, 3.66643801060135496247185274273, 4.66125794347334975754976010290, 5.34295079124919823511844933757, 5.93360966412049476443514792853, 6.70438184457472864541167980056, 7.55071437538943836623405220284

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