Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.46·5-s + 9-s + 1.46·11-s + 2·13-s + 3.46·15-s − 0.535·17-s + 6.92·19-s − 1.46·23-s + 6.99·25-s + 27-s + 4.92·29-s + 10.9·31-s + 1.46·33-s + 2·37-s + 2·39-s − 11.4·41-s + 8·43-s + 3.46·45-s − 10.9·47-s − 0.535·51-s + 2·53-s + 5.07·55-s + 6.92·57-s − 1.07·59-s − 8.92·61-s + 6.92·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.54·5-s + 0.333·9-s + 0.441·11-s + 0.554·13-s + 0.894·15-s − 0.129·17-s + 1.58·19-s − 0.305·23-s + 1.39·25-s + 0.192·27-s + 0.915·29-s + 1.96·31-s + 0.254·33-s + 0.328·37-s + 0.320·39-s − 1.79·41-s + 1.21·43-s + 0.516·45-s − 1.59·47-s − 0.0750·51-s + 0.274·53-s + 0.683·55-s + 0.917·57-s − 0.139·59-s − 1.14·61-s + 0.859·65-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  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.430548955\)
\(L(\frac12)\)  \(\approx\)  \(4.430548955\)
\(L(\frac{3}{2})\)   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 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 0.535T + 17T^{2} \)
19 \( 1 - 6.92T + 19T^{2} \)
23 \( 1 + 1.46T + 23T^{2} \)
29 \( 1 - 4.92T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 11.4T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.07T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 - 8.92T + 97T^{2} \)
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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

−7.81752263130246188765490064504, −6.81465441514980751787628056285, −6.41475120588702610241165444292, −5.70223318803946767535037767441, −5.00919488270935039219713240695, −4.23773742306480475982478718798, −3.16220555108758974624089982498, −2.70174786262735373283221458002, −1.65944002383093194328141439287, −1.10128092783868310464318061590, 1.10128092783868310464318061590, 1.65944002383093194328141439287, 2.70174786262735373283221458002, 3.16220555108758974624089982498, 4.23773742306480475982478718798, 5.00919488270935039219713240695, 5.70223318803946767535037767441, 6.41475120588702610241165444292, 6.81465441514980751787628056285, 7.81752263130246188765490064504

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