Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \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  − 3.27·5-s + 3.27·11-s + 6.27·13-s + 4·17-s + 6.27·19-s + 4·23-s + 5.72·25-s − 5.27·29-s + 31-s − 2.27·37-s + 4.54·41-s − 0.274·43-s − 6·47-s − 9.27·53-s − 10.7·55-s − 1.27·59-s + 10·61-s − 20.5·65-s + 0.274·67-s + 2·71-s − 4.27·73-s − 11.5·79-s + 7.27·83-s − 13.0·85-s + 10.5·89-s − 20.5·95-s + 8.72·97-s + ⋯
L(s)  = 1  − 1.46·5-s + 0.987·11-s + 1.74·13-s + 0.970·17-s + 1.43·19-s + 0.834·23-s + 1.14·25-s − 0.979·29-s + 0.179·31-s − 0.373·37-s + 0.710·41-s − 0.0419·43-s − 0.875·47-s − 1.27·53-s − 1.44·55-s − 0.165·59-s + 1.28·61-s − 2.54·65-s + 0.0335·67-s + 0.237·71-s − 0.500·73-s − 1.29·79-s + 0.798·83-s − 1.42·85-s + 1.11·89-s − 2.10·95-s + 0.885·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7056} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.948650647\)
\(L(\frac12)\)  \(\approx\)  \(1.948650647\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.27T + 5T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 1.27T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 0.274T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 4.27T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−7.86978770103416602588758663709, −7.38432096828305060550012975114, −6.63162347293335634661149273941, −5.87111459646461488024548039744, −5.06766336960376844669487522255, −4.13857392682189700293555059576, −3.51135275609438919697679605344, −3.19439580237188933494525050552, −1.48557238731539911828864422295, −0.795649439728604679389783437840, 0.795649439728604679389783437840, 1.48557238731539911828864422295, 3.19439580237188933494525050552, 3.51135275609438919697679605344, 4.13857392682189700293555059576, 5.06766336960376844669487522255, 5.87111459646461488024548039744, 6.63162347293335634661149273941, 7.38432096828305060550012975114, 7.86978770103416602588758663709

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