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 − 2.82·5-s + 9-s + 2.82·11-s + 2.82·15-s − 2.82·17-s − 4·19-s + 8.48·23-s + 3.00·25-s − 27-s − 2·29-s − 2.82·33-s + 6·37-s − 8.48·41-s − 11.3·43-s − 2.82·45-s + 8·47-s + 2.82·51-s − 6·53-s − 8.00·55-s + 4·57-s − 12·59-s + 5.65·61-s + 5.65·67-s − 8.48·69-s + 2.82·71-s + 5.65·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.26·5-s + 0.333·9-s + 0.852·11-s + 0.730·15-s − 0.685·17-s − 0.917·19-s + 1.76·23-s + 0.600·25-s − 0.192·27-s − 0.371·29-s − 0.492·33-s + 0.986·37-s − 1.32·41-s − 1.72·43-s − 0.421·45-s + 1.16·47-s + 0.396·51-s − 0.824·53-s − 1.07·55-s + 0.529·57-s − 1.56·59-s + 0.724·61-s + 0.691·67-s − 1.02·69-s + 0.335·71-s + 0.662·73-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  =  \(1\)
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 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 - 2.82T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 8.48T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 2.82T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−7.32183936039722784415471531742, −6.61757862339356182139985613531, −6.26258573794670946895669074931, −5.04008817464028213452904902084, −4.66121016363799056194743154946, −3.83666809803071275803343039271, −3.30462262386465517806987162958, −2.10214034970733727892519418987, −0.999600496970624257803597758027, 0, 0.999600496970624257803597758027, 2.10214034970733727892519418987, 3.30462262386465517806987162958, 3.83666809803071275803343039271, 4.66121016363799056194743154946, 5.04008817464028213452904902084, 6.26258573794670946895669074931, 6.61757862339356182139985613531, 7.32183936039722784415471531742

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