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 + 4·5-s + 9-s + 6·11-s − 5·13-s − 4·15-s + 2·17-s + 19-s + 6·23-s + 11·25-s − 27-s + 3·31-s − 6·33-s − 3·37-s + 5·39-s − 6·41-s + 5·43-s + 4·45-s + 4·47-s − 2·51-s + 6·53-s + 24·55-s − 57-s − 6·59-s + 2·61-s − 20·65-s + 7·67-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 1/3·9-s + 1.80·11-s − 1.38·13-s − 1.03·15-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s − 0.192·27-s + 0.538·31-s − 1.04·33-s − 0.493·37-s + 0.800·39-s − 0.937·41-s + 0.762·43-s + 0.596·45-s + 0.583·47-s − 0.280·51-s + 0.824·53-s + 3.23·55-s − 0.132·57-s − 0.781·59-s + 0.256·61-s − 2.48·65-s + 0.855·67-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  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.008024687\)
\(L(\frac12)\)  \(\approx\)  \(3.008024687\)
\(L(\frac{3}{2})\)   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 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.31028868468344221314029262729, −6.95494016934533717229271227600, −6.26185881241052233801059647454, −5.72560251251270824439958695398, −5.04643913560879425104597015367, −4.47406223049290050766796287891, −3.34666905180181432452181232543, −2.46536529994710078338480153790, −1.61712913086541090433107997272, −0.932245181812917662550156814272, 0.932245181812917662550156814272, 1.61712913086541090433107997272, 2.46536529994710078338480153790, 3.34666905180181432452181232543, 4.47406223049290050766796287891, 5.04643913560879425104597015367, 5.72560251251270824439958695398, 6.26185881241052233801059647454, 6.95494016934533717229271227600, 7.31028868468344221314029262729

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