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 − 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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
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

−13.30229196428434, −13.14767089399178, −12.76524027913840, −12.18158614063132, −11.50230240860387, −11.16824687630749, −10.92126121377260, −10.18277100919119, −9.667463077855282, −9.273905051977257, −8.650285159009959, −8.258987804729182, −7.573492821127069, −6.938925744933210, −6.575279646533036, −6.309693773645356, −5.513060475377837, −5.118642076519572, −4.524228379404657, −3.884291624514350, −3.553448533869142, −2.805800614920292, −2.401932368020356, −1.649556713716323, −0.8726571037675081, 0, 0.8726571037675081, 1.649556713716323, 2.401932368020356, 2.805800614920292, 3.553448533869142, 3.884291624514350, 4.524228379404657, 5.118642076519572, 5.513060475377837, 6.309693773645356, 6.575279646533036, 6.938925744933210, 7.573492821127069, 8.258987804729182, 8.650285159009959, 9.273905051977257, 9.667463077855282, 10.18277100919119, 10.92126121377260, 11.16824687630749, 11.50230240860387, 12.18158614063132, 12.76524027913840, 13.14767089399178, 13.30229196428434

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