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 − 5-s + 9-s − 5·11-s − 15-s − 4·17-s − 8·19-s − 4·23-s − 4·25-s + 27-s + 5·29-s + 3·31-s − 5·33-s + 4·37-s − 2·43-s − 45-s − 6·47-s − 4·51-s + 9·53-s + 5·55-s − 8·57-s + 11·59-s + 6·61-s + 2·67-s − 4·69-s + 2·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s − 0.258·15-s − 0.970·17-s − 1.83·19-s − 0.834·23-s − 4/5·25-s + 0.192·27-s + 0.928·29-s + 0.538·31-s − 0.870·33-s + 0.657·37-s − 0.304·43-s − 0.149·45-s − 0.875·47-s − 0.560·51-s + 1.23·53-s + 0.674·55-s − 1.05·57-s + 1.43·59-s + 0.768·61-s + 0.244·67-s − 0.481·69-s + 0.237·71-s + 1.17·73-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

\( d \)  =  \(2\)
\( N \)  =  \(9408\)    =    \(2^{6} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{9408} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.292591498\)
\(L(\frac12)\)  \(\approx\)  \(1.292591498\)
\(L(\frac{3}{2})\)   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 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−7.998681053745703793364366877448, −7.04366644584198172189575062898, −6.45981038732264420135306629742, −5.66650174113303225303400727510, −4.72879903370893444910528997607, −4.24383474579913416545921421180, −3.45991871456905810621507852744, −2.36451899056830954596380463677, −2.17031572588683744082763631571, −0.49283329760408682331238590774, 0.49283329760408682331238590774, 2.17031572588683744082763631571, 2.36451899056830954596380463677, 3.45991871456905810621507852744, 4.24383474579913416545921421180, 4.72879903370893444910528997607, 5.66650174113303225303400727510, 6.45981038732264420135306629742, 7.04366644584198172189575062898, 7.998681053745703793364366877448

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