Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17 $
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 − 2·7-s + 9-s − 11-s − 4·13-s − 17-s + 2·19-s + 2·21-s + 2·23-s − 27-s + 6·29-s − 8·31-s + 33-s − 2·37-s + 4·39-s − 2·41-s − 2·43-s + 4·47-s − 3·49-s + 51-s − 12·53-s − 2·57-s + 10·59-s + 6·61-s − 2·63-s − 12·67-s − 2·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.242·17-s + 0.458·19-s + 0.436·21-s + 0.417·23-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s − 0.328·37-s + 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.583·47-s − 3/7·49-s + 0.140·51-s − 1.64·53-s − 0.264·57-s + 1.30·59-s + 0.768·61-s − 0.251·63-s − 1.46·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{224400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224400,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.2689229006\)
\(L(\frac12)\)  \(\approx\)  \(0.2689229006\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−12.82128851313705, −12.52211888528544, −12.10309962958936, −11.52575088391155, −11.18646902482400, −10.55002971034605, −10.13533544654807, −9.810770221605099, −9.263606832974230, −8.833904259487513, −8.212577182743507, −7.587418516204489, −7.187496931452813, −6.751529095219015, −6.330336276960831, −5.522820501632780, −5.412109265478411, −4.638651232825383, −4.341517144253035, −3.461211143218900, −3.070584431844001, −2.475262974145288, −1.782166100447225, −1.070283622276284, −0.1610506129749744, 0.1610506129749744, 1.070283622276284, 1.782166100447225, 2.475262974145288, 3.070584431844001, 3.461211143218900, 4.341517144253035, 4.638651232825383, 5.412109265478411, 5.522820501632780, 6.330336276960831, 6.751529095219015, 7.187496931452813, 7.587418516204489, 8.212577182743507, 8.833904259487513, 9.263606832974230, 9.810770221605099, 10.13533544654807, 10.55002971034605, 11.18646902482400, 11.52575088391155, 12.10309962958936, 12.52211888528544, 12.82128851313705

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