Properties

Degree $2$
Conductor $44400$
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 + 9-s − 4·11-s + 2·13-s − 2·17-s − 4·19-s − 8·23-s − 27-s − 2·29-s − 8·31-s + 4·33-s − 37-s − 2·39-s + 10·41-s + 12·43-s − 7·49-s + 2·51-s − 6·53-s + 4·57-s − 4·59-s − 10·61-s − 4·67-s + 8·69-s − 8·71-s + 6·73-s + 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.164·37-s − 0.320·39-s + 1.56·41-s + 1.82·43-s − 49-s + 0.280·51-s − 0.824·53-s + 0.529·57-s − 0.520·59-s − 1.28·61-s − 0.488·67-s + 0.963·69-s − 0.949·71-s + 0.702·73-s + 0.900·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{44400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 44400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3534144891\)
\(L(\frac12)\) \(\approx\) \(0.3534144891\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
37 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.69796694865100, −14.08177864475589, −13.60015437445668, −13.00795699185803, −12.52051892084466, −12.31643278748361, −11.33787279410287, −10.98014262458064, −10.65550528613966, −10.10739217117142, −9.331940500769041, −9.013173194780919, −8.108065296960344, −7.787685491242534, −7.267071676592791, −6.377652604159069, −6.015428295387967, −5.550663072172700, −4.806314437020335, −4.203956459247198, −3.718409059429040, −2.740025267487433, −2.141350717168606, −1.424968442595003, −0.2159638353039421, 0.2159638353039421, 1.424968442595003, 2.141350717168606, 2.740025267487433, 3.718409059429040, 4.203956459247198, 4.806314437020335, 5.550663072172700, 6.015428295387967, 6.377652604159069, 7.267071676592791, 7.787685491242534, 8.108065296960344, 9.013173194780919, 9.331940500769041, 10.10739217117142, 10.65550528613966, 10.98014262458064, 11.33787279410287, 12.31643278748361, 12.52051892084466, 13.00795699185803, 13.60015437445668, 14.08177864475589, 14.69796694865100

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