Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11 $
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 + 7-s + 9-s + 11-s − 6·13-s − 2·17-s − 4·19-s − 21-s − 27-s − 2·29-s − 8·31-s − 33-s − 6·37-s + 6·39-s + 10·41-s − 4·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s + 63-s − 12·67-s − 2·73-s + 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.986·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.125·63-s − 1.46·67-s − 0.234·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(92400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{92400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 92400,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.53427715820791, −13.88208911563660, −13.13064375793251, −12.89766146549001, −12.18403243248220, −12.04737976862359, −11.33049742597702, −10.89812299500367, −10.48061194153566, −9.939113767715422, −9.211955268219392, −9.103590985471667, −8.272086466471085, −7.585738478306445, −7.366652077930470, −6.687246081102188, −6.206679440049020, −5.587947688174274, −4.979383876846096, −4.604151452263081, −4.066804855634753, −3.297815278093934, −2.544923328362849, −1.905137864263962, −1.383250620206822, 0, 0, 1.383250620206822, 1.905137864263962, 2.544923328362849, 3.297815278093934, 4.066804855634753, 4.604151452263081, 4.979383876846096, 5.587947688174274, 6.206679440049020, 6.687246081102188, 7.366652077930470, 7.585738478306445, 8.272086466471085, 9.103590985471667, 9.211955268219392, 9.939113767715422, 10.48061194153566, 10.89812299500367, 11.33049742597702, 12.04737976862359, 12.18403243248220, 12.89766146549001, 13.13064375793251, 13.88208911563660, 14.53427715820791

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