Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
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 + 6·11-s + 2·13-s − 4·19-s − 6·23-s − 5·25-s + 27-s − 6·29-s − 8·31-s + 6·33-s − 2·37-s + 2·39-s − 12·41-s + 4·43-s − 12·47-s + 6·53-s − 4·57-s − 10·61-s − 8·67-s − 6·69-s + 6·71-s + 10·73-s − 5·75-s − 4·79-s + 81-s − 12·83-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s + 1.80·11-s + 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s − 1.87·41-s + 0.609·43-s − 1.75·47-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.977·67-s − 0.722·69-s + 0.712·71-s + 1.17·73-s − 0.577·75-s − 0.450·79-s + 1/9·81-s − 1.31·83-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

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.33978744208638478150447194311, −6.68018359677214897577974504311, −6.12118933536028124716817869873, −5.38931467471481918720560102508, −4.17509149223426232107219639461, −3.92899224204306789996334105668, −3.22673417481384035100368040671, −1.84987696198250663319067290149, −1.64262388040949769268493081981, 0, 1.64262388040949769268493081981, 1.84987696198250663319067290149, 3.22673417481384035100368040671, 3.92899224204306789996334105668, 4.17509149223426232107219639461, 5.38931467471481918720560102508, 6.12118933536028124716817869873, 6.68018359677214897577974504311, 7.33978744208638478150447194311

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