Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4·7-s + 8-s − 6·11-s − 4·13-s + 4·14-s + 16-s + 6·17-s − 6·22-s + 6·23-s − 4·26-s + 4·28-s + 2·29-s + 32-s + 6·34-s − 8·37-s + 10·41-s + 4·43-s − 6·44-s + 6·46-s − 2·47-s + 9·49-s − 4·52-s − 10·53-s + 4·56-s + 2·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s − 1.80·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 1.27·22-s + 1.25·23-s − 0.784·26-s + 0.755·28-s + 0.371·29-s + 0.176·32-s + 1.02·34-s − 1.31·37-s + 1.56·41-s + 0.609·43-s − 0.904·44-s + 0.884·46-s − 0.291·47-s + 9/7·49-s − 0.554·52-s − 1.37·53-s + 0.534·56-s + 0.262·58-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: \(1\)
Selberg data: \((2,\ 162450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−13.58958214522438, −12.90201073033472, −12.60250400790024, −12.12169194694095, −11.77773710571374, −10.92724253666885, −10.83365011442319, −10.44202278568791, −9.708014844869407, −9.311906705764995, −8.441318197562860, −8.044904544074796, −7.608266623679952, −7.389491889724642, −6.741285481156611, −5.819619272470075, −5.466909445423959, −5.040351610363922, −4.725244608568273, −4.203678551500717, −3.200630825011197, −2.892582412253408, −2.327082782493521, −1.630177611195930, −1.016228742583560, 0, 1.016228742583560, 1.630177611195930, 2.327082782493521, 2.892582412253408, 3.200630825011197, 4.203678551500717, 4.725244608568273, 5.040351610363922, 5.466909445423959, 5.819619272470075, 6.741285481156611, 7.389491889724642, 7.608266623679952, 8.044904544074796, 8.441318197562860, 9.311906705764995, 9.708014844869407, 10.44202278568791, 10.83365011442319, 10.92724253666885, 11.77773710571374, 12.12169194694095, 12.60250400790024, 12.90201073033472, 13.58958214522438

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