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 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5·7-s + 9-s + 11-s − 6·13-s + 17-s + 2·19-s + 5·21-s + 6·23-s − 27-s − 7·29-s − 33-s − 8·37-s + 6·39-s − 7·41-s − 4·43-s + 7·47-s + 18·49-s − 51-s − 7·53-s − 2·57-s − 9·59-s + 14·61-s − 5·63-s − 3·67-s − 6·69-s + 73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.242·17-s + 0.458·19-s + 1.09·21-s + 1.25·23-s − 0.192·27-s − 1.29·29-s − 0.174·33-s − 1.31·37-s + 0.960·39-s − 1.09·41-s − 0.609·43-s + 1.02·47-s + 18/7·49-s − 0.140·51-s − 0.961·53-s − 0.264·57-s − 1.17·59-s + 1.79·61-s − 0.629·63-s − 0.366·67-s − 0.722·69-s + 0.117·73-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  =  \(2\)
Selberg data  =  \((2,\ 224400,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.19463434629206, −12.88501338480820, −12.50105447210611, −12.12232182236153, −11.64418272450190, −11.16405686026909, −10.47033128898471, −10.04134696593179, −9.846668293529434, −9.164220816463807, −9.072196575776294, −8.249022414264977, −7.390732587300792, −7.163507466307661, −6.849574666783103, −6.307557437405131, −5.640579056233936, −5.341009884028344, −4.785214652887581, −4.111780458647538, −3.455739365323263, −3.123049762203356, −2.541730633868772, −1.793865031348323, −0.9987867938234462, 0, 0, 0.9987867938234462, 1.793865031348323, 2.541730633868772, 3.123049762203356, 3.455739365323263, 4.111780458647538, 4.785214652887581, 5.341009884028344, 5.640579056233936, 6.307557437405131, 6.849574666783103, 7.163507466307661, 7.390732587300792, 8.249022414264977, 9.072196575776294, 9.164220816463807, 9.846668293529434, 10.04134696593179, 10.47033128898471, 11.16405686026909, 11.64418272450190, 12.12232182236153, 12.50105447210611, 12.88501338480820, 13.19463434629206

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