Properties

Degree $2$
Conductor $162450$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s − 6·17-s + 6·23-s + 2·26-s − 2·28-s + 4·29-s − 32-s + 6·34-s − 10·37-s + 8·41-s − 2·43-s − 6·46-s − 2·47-s − 3·49-s − 2·52-s − 2·53-s + 2·56-s − 4·58-s + 14·61-s + 64-s − 4·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.25·23-s + 0.392·26-s − 0.377·28-s + 0.742·29-s − 0.176·32-s + 1.02·34-s − 1.64·37-s + 1.24·41-s − 0.304·43-s − 0.884·46-s − 0.291·47-s − 3/7·49-s − 0.277·52-s − 0.274·53-s + 0.267·56-s − 0.525·58-s + 1.79·61-s + 1/8·64-s − 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(162450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{162450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 162450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5822940479\)
\(L(\frac12)\) \(\approx\) \(0.5822940479\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−13.22920054681147, −12.71399347026878, −12.41360493748835, −11.60304481098924, −11.44876206994834, −10.77691899515348, −10.29642366135921, −10.02352260074546, −9.260680185262224, −9.026817317968174, −8.619035279948606, −7.999029124810822, −7.371666856324210, −6.870638757712823, −6.665243000209890, −6.045365516140104, −5.393858502355934, −4.801469293463051, −4.310509299600308, −3.533875078311316, −2.969791780497106, −2.500119493718730, −1.851706124870551, −1.091032707431199, −0.2684379637223729, 0.2684379637223729, 1.091032707431199, 1.851706124870551, 2.500119493718730, 2.969791780497106, 3.533875078311316, 4.310509299600308, 4.801469293463051, 5.393858502355934, 6.045365516140104, 6.665243000209890, 6.870638757712823, 7.371666856324210, 7.999029124810822, 8.619035279948606, 9.026817317968174, 9.260680185262224, 10.02352260074546, 10.29642366135921, 10.77691899515348, 11.44876206994834, 11.60304481098924, 12.41360493748835, 12.71399347026878, 13.22920054681147

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