Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5 \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 + 5-s + 9-s + 2·13-s − 15-s + 6·17-s − 8·19-s + 25-s − 27-s − 6·29-s − 4·31-s + 10·37-s − 2·39-s + 6·41-s − 4·43-s + 45-s − 6·51-s + 6·53-s + 8·57-s + 12·59-s − 10·61-s + 2·65-s − 4·67-s − 12·71-s + 10·73-s − 75-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s − 0.258·15-s + 1.45·17-s − 1.83·19-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.149·45-s − 0.840·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.115·75-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(47040\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{47040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 47040,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.886858607\)
\(L(\frac12)\)  \(\approx\)  \(1.886858607\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
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 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.71255790774126, −14.20280687141894, −13.32156467875001, −13.03314449810014, −12.67885713248348, −11.98101319339784, −11.47853600936032, −10.90996660398930, −10.52577390229715, −9.944118315716642, −9.476602520026193, −8.795829380901138, −8.339583782811815, −7.548853412518320, −7.227568973120985, −6.287213434554290, −6.020559162066488, −5.559061119662327, −4.830425887845511, −4.142259969927957, −3.664421837666365, −2.790562818768234, −2.037439524964905, −1.367112172762159, −0.5242706823290154, 0.5242706823290154, 1.367112172762159, 2.037439524964905, 2.790562818768234, 3.664421837666365, 4.142259969927957, 4.830425887845511, 5.559061119662327, 6.020559162066488, 6.287213434554290, 7.227568973120985, 7.548853412518320, 8.339583782811815, 8.795829380901138, 9.476602520026193, 9.944118315716642, 10.52577390229715, 10.90996660398930, 11.47853600936032, 11.98101319339784, 12.67885713248348, 13.03314449810014, 13.32156467875001, 14.20280687141894, 14.71255790774126

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