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  − 2·5-s + 4·11-s + 2·13-s − 6·17-s + 4·19-s − 25-s + 2·29-s + 6·37-s + 2·41-s + 4·43-s − 6·53-s − 8·55-s − 12·59-s + 2·61-s − 4·65-s − 4·67-s + 6·73-s + 16·79-s + 12·83-s + 12·85-s − 14·89-s − 8·95-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 0.986·37-s + 0.312·41-s + 0.609·43-s − 0.824·53-s − 1.07·55-s − 1.56·59-s + 0.256·61-s − 0.496·65-s − 0.488·67-s + 0.702·73-s + 1.80·79-s + 1.31·83-s + 1.30·85-s − 1.48·89-s − 0.820·95-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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  :  $\chi_{7056} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7056,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.668048632\)
\(L(\frac12)\)  \(\approx\)  \(1.668048632\)
\(L(\frac{3}{2})\)   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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−17.17533746204532, −16.48020446926384, −15.98380828216118, −15.40701567769956, −15.00271713102104, −14.11279568911430, −13.76183764821975, −13.01358366727710, −12.23988800550615, −11.81927054067745, −11.08063411545196, −10.91295476656703, −9.692669203954028, −9.261436054727305, −8.606114381489563, −7.906821987114439, −7.300735279584038, −6.498043514045001, −6.063709323995312, −4.931848512241406, −4.229461094327008, −3.729330108104265, −2.838766017044096, −1.709389925981436, −0.6739217867645482, 0.6739217867645482, 1.709389925981436, 2.838766017044096, 3.729330108104265, 4.229461094327008, 4.931848512241406, 6.063709323995312, 6.498043514045001, 7.300735279584038, 7.906821987114439, 8.606114381489563, 9.261436054727305, 9.692669203954028, 10.91295476656703, 11.08063411545196, 11.81927054067745, 12.23988800550615, 13.01358366727710, 13.76183764821975, 14.11279568911430, 15.00271713102104, 15.40701567769956, 15.98380828216118, 16.48020446926384, 17.17533746204532

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