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  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.419875652\)
\(L(\frac12)\)  \(\approx\)  \(3.419875652\)
\(L(\frac{3}{2})\)   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

−7.73421539395236366430001804493, −6.96541645915432855218597929462, −6.34679548015235741300450429702, −5.86016148269565957058685991509, −4.76714180035995561453715019228, −4.15734441221853704248790189745, −3.46995114737342128672316311526, −2.57126246579998573602374877697, −1.76638690175158197459294636409, −0.907169062729814406464088705719, 0.907169062729814406464088705719, 1.76638690175158197459294636409, 2.57126246579998573602374877697, 3.46995114737342128672316311526, 4.15734441221853704248790189745, 4.76714180035995561453715019228, 5.86016148269565957058685991509, 6.34679548015235741300450429702, 6.96541645915432855218597929462, 7.73421539395236366430001804493

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