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 − 4·7-s + 9-s + 11-s + 3·13-s + 17-s − 19-s + 4·21-s − 23-s − 27-s + 7·29-s + 9·31-s − 33-s + 4·37-s − 3·39-s − 6·41-s − 11·43-s + 9·49-s − 51-s + 6·53-s + 57-s − 4·59-s − 6·61-s − 4·63-s + 14·67-s + 69-s + 71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.242·17-s − 0.229·19-s + 0.872·21-s − 0.208·23-s − 0.192·27-s + 1.29·29-s + 1.61·31-s − 0.174·33-s + 0.657·37-s − 0.480·39-s − 0.937·41-s − 1.67·43-s + 9/7·49-s − 0.140·51-s + 0.824·53-s + 0.132·57-s − 0.520·59-s − 0.768·61-s − 0.503·63-s + 1.71·67-s + 0.120·69-s + 0.118·71-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  =  \((2,\ 224400,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.973586381\)
\(L(\frac12)\)  \(\approx\)  \(1.973586381\)
\(L(\frac{3}{2})\)   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−12.94587280263770, −12.45636909588239, −11.98303718570389, −11.72522867265497, −11.09283686534294, −10.53389909151602, −10.12797800498197, −9.789554710734174, −9.374967271953949, −8.606476311054103, −8.384135459519772, −7.750616920397640, −7.012617767995740, −6.564923552067340, −6.265217344346437, −6.041971816725649, −5.111536588995405, −4.819599953583409, −4.044165174985211, −3.568417065314183, −3.135763871705831, −2.505672467222931, −1.745961126591687, −0.8905491558930344, −0.5223635544899653, 0.5223635544899653, 0.8905491558930344, 1.745961126591687, 2.505672467222931, 3.135763871705831, 3.568417065314183, 4.044165174985211, 4.819599953583409, 5.111536588995405, 6.041971816725649, 6.265217344346437, 6.564923552067340, 7.012617767995740, 7.750616920397640, 8.384135459519772, 8.606476311054103, 9.374967271953949, 9.789554710734174, 10.12797800498197, 10.53389909151602, 11.09283686534294, 11.72522867265497, 11.98303718570389, 12.45636909588239, 12.94587280263770

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