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  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.564166087\)
\(L(\frac12)\)  \(\approx\)  \(1.564166087\)
\(L(\frac{3}{2})\)   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

−7.66704391563267466408747033453, −7.15916233678585242896748487052, −6.07186208611442724160614588288, −5.63572265312026843734928392292, −5.17949058979060087276286358320, −4.34753614684461177561158872944, −3.33263639061551217851718650296, −2.65508443181212071696941839887, −1.65045674899787460442941380444, −0.62514295325464923881788964725, 0.62514295325464923881788964725, 1.65045674899787460442941380444, 2.65508443181212071696941839887, 3.33263639061551217851718650296, 4.34753614684461177561158872944, 5.17949058979060087276286358320, 5.63572265312026843734928392292, 6.07186208611442724160614588288, 7.15916233678585242896748487052, 7.66704391563267466408747033453

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