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 − 2·5-s + 9-s + 6·11-s + 3·13-s − 2·15-s + 4·17-s + 5·19-s − 4·23-s − 25-s + 27-s + 4·29-s + 7·31-s + 6·33-s + 9·37-s + 3·39-s − 2·41-s + 43-s − 2·45-s + 2·47-s + 4·51-s − 8·53-s − 12·55-s + 5·57-s − 10·61-s − 6·65-s + 15·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.80·11-s + 0.832·13-s − 0.516·15-s + 0.970·17-s + 1.14·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.25·31-s + 1.04·33-s + 1.47·37-s + 0.480·39-s − 0.312·41-s + 0.152·43-s − 0.298·45-s + 0.291·47-s + 0.560·51-s − 1.09·53-s − 1.61·55-s + 0.662·57-s − 1.28·61-s − 0.744·65-s + 1.83·67-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  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.005979036\)
\(L(\frac12)\)  \(\approx\)  \(3.005979036\)
\(L(\frac{3}{2})\)   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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.79135202024277334389639114403, −7.15922049480210701139319720327, −6.32854729307507255571323398252, −5.86405387253474759787898954785, −4.64101075926790658806218447437, −4.07494342133225173372309331792, −3.50933640798374715298025226500, −2.86472456216643268078313135158, −1.51892061640086881884189679623, −0.909468114889031403038423095373, 0.909468114889031403038423095373, 1.51892061640086881884189679623, 2.86472456216643268078313135158, 3.50933640798374715298025226500, 4.07494342133225173372309331792, 4.64101075926790658806218447437, 5.86405387253474759787898954785, 6.32854729307507255571323398252, 7.15922049480210701139319720327, 7.79135202024277334389639114403

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