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 + 2.82·5-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  =  \(0\)
Selberg data  =  \((2,\ 9408,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.256313281\)
\(L(\frac12)\)  \(\approx\)  \(4.256313281\)
\(L(\frac{3}{2})\)   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

−7.72422264843946343694557146770, −6.91027350984720031143776283548, −6.45938602286487460470251211713, −5.56338792392088227219697612548, −5.11148135797121067663999258614, −4.14866656069026237825313021656, −3.24941329787843239880586863394, −2.68967747548855150321070991231, −1.64808447400130624722306976293, −1.07740963425290224405502400682, 1.07740963425290224405502400682, 1.64808447400130624722306976293, 2.68967747548855150321070991231, 3.24941329787843239880586863394, 4.14866656069026237825313021656, 5.11148135797121067663999258614, 5.56338792392088227219697612548, 6.45938602286487460470251211713, 6.91027350984720031143776283548, 7.72422264843946343694557146770

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