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 + 3·5-s + 9-s − 11-s − 4·13-s − 3·15-s − 4·17-s − 8·23-s + 4·25-s − 27-s + 7·29-s + 11·31-s + 33-s − 4·37-s + 4·39-s + 4·41-s − 2·43-s + 3·45-s − 2·47-s + 4·51-s + 11·53-s − 3·55-s − 7·59-s + 10·61-s − 12·65-s + 10·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.774·15-s − 0.970·17-s − 1.66·23-s + 4/5·25-s − 0.192·27-s + 1.29·29-s + 1.97·31-s + 0.174·33-s − 0.657·37-s + 0.640·39-s + 0.624·41-s − 0.304·43-s + 0.447·45-s − 0.291·47-s + 0.560·51-s + 1.51·53-s − 0.404·55-s − 0.911·59-s + 1.28·61-s − 1.48·65-s + 1.22·67-s + 0.963·69-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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 11 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.12620817674784267361460830161, −6.56361543604964062804729157647, −6.02513860435056297159148063767, −5.35421106503991827982959113732, −4.72372977064486876419593012710, −4.06029876054638294059472301093, −2.62533600786733137152708090652, −2.32024691446897912779845960385, −1.26586878960909953859973408971, 0, 1.26586878960909953859973408971, 2.32024691446897912779845960385, 2.62533600786733137152708090652, 4.06029876054638294059472301093, 4.72372977064486876419593012710, 5.35421106503991827982959113732, 6.02513860435056297159148063767, 6.56361543604964062804729157647, 7.12620817674784267361460830161

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