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 + 9-s − 2·11-s − 2·13-s − 4·17-s + 4·19-s + 6·23-s − 5·25-s + 27-s + 2·29-s − 2·33-s + 6·37-s − 2·39-s − 8·41-s − 8·43-s − 4·47-s − 4·51-s + 6·53-s + 4·57-s − 14·61-s + 4·67-s + 6·69-s + 2·71-s + 2·73-s − 5·75-s − 4·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.917·19-s + 1.25·23-s − 25-s + 0.192·27-s + 0.371·29-s − 0.348·33-s + 0.986·37-s − 0.320·39-s − 1.24·41-s − 1.21·43-s − 0.583·47-s − 0.560·51-s + 0.824·53-s + 0.529·57-s − 1.79·61-s + 0.488·67-s + 0.722·69-s + 0.237·71-s + 0.234·73-s − 0.577·75-s − 0.450·79-s + 1/9·81-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.41316834474548691555357375667, −6.83364278896222152523375167060, −6.04976907103984301589697009940, −5.08821231226396920453742081831, −4.71093761217536498900331735499, −3.71043880881469772089910145016, −2.96909066521507484542218671027, −2.31698807390248258603905073049, −1.34156614571833066868188465618, 0, 1.34156614571833066868188465618, 2.31698807390248258603905073049, 2.96909066521507484542218671027, 3.71043880881469772089910145016, 4.71093761217536498900331735499, 5.08821231226396920453742081831, 6.04976907103984301589697009940, 6.83364278896222152523375167060, 7.41316834474548691555357375667

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