Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.41·5-s + 9-s + 2·11-s − 2.58·13-s + 3.41·15-s − 2.24·17-s + 2.82·19-s − 7.65·23-s + 6.65·25-s − 27-s + 6.82·29-s − 1.17·31-s − 2·33-s + 4·37-s + 2.58·39-s + 6.24·41-s − 5.65·43-s − 3.41·45-s − 2.82·47-s + 2.24·51-s + 2·53-s − 6.82·55-s − 2.82·57-s + 1.17·59-s − 12.2·61-s + 8.82·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.52·5-s + 0.333·9-s + 0.603·11-s − 0.717·13-s + 0.881·15-s − 0.543·17-s + 0.648·19-s − 1.59·23-s + 1.33·25-s − 0.192·27-s + 1.26·29-s − 0.210·31-s − 0.348·33-s + 0.657·37-s + 0.414·39-s + 0.974·41-s − 0.862·43-s − 0.508·45-s − 0.412·47-s + 0.314·51-s + 0.274·53-s − 0.920·55-s − 0.374·57-s + 0.152·59-s − 1.56·61-s + 1.09·65-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  =  \(1\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + 3.41T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 6.24T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 1.17T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 9.31T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 7.31T + 83T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 - 2.58T + 97T^{2} \)
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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

−7.37926055663323740349445626231, −6.72179868179206086224145312982, −6.12698372246636125989589359703, −5.15438545557841502301353931146, −4.47709280091476783126997238757, −3.99191189864677436049537304978, −3.21949935623726062744663923486, −2.17419224357951494962394437670, −0.925649264892962053697896084650, 0, 0.925649264892962053697896084650, 2.17419224357951494962394437670, 3.21949935623726062744663923486, 3.99191189864677436049537304978, 4.47709280091476783126997238757, 5.15438545557841502301353931146, 6.12698372246636125989589359703, 6.72179868179206086224145312982, 7.37926055663323740349445626231

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