Properties

Degree $2$
Conductor $176400$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s + 2·13-s + 4·17-s − 8·23-s − 2·31-s − 8·37-s − 2·41-s − 2·43-s + 10·47-s − 2·53-s − 4·59-s + 10·61-s + 2·67-s − 12·71-s + 10·73-s − 16·79-s + 16·83-s + 14·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.66·23-s − 0.359·31-s − 1.31·37-s − 0.312·41-s − 0.304·43-s + 1.45·47-s − 0.274·53-s − 0.520·59-s + 1.28·61-s + 0.244·67-s − 1.42·71-s + 1.17·73-s − 1.80·79-s + 1.75·83-s + 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{176400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.213356018\)
\(L(\frac12)\) \(\approx\) \(2.213356018\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.22216015109653, −12.64660971198113, −12.11152099088996, −11.88064543471111, −11.42403267706662, −10.71616089464343, −10.28263640533346, −10.02758741544144, −9.244948603106116, −8.982002846163236, −8.390930751673256, −7.823308242891913, −7.552899562667763, −6.714204986471061, −6.480596833828254, −5.731102940099073, −5.482989704200729, −4.793316890450383, −4.015197241347150, −3.769911024965562, −3.213833168171274, −2.422841196608842, −1.781387448962806, −1.261018907688362, −0.4391615337022185, 0.4391615337022185, 1.261018907688362, 1.781387448962806, 2.422841196608842, 3.213833168171274, 3.769911024965562, 4.015197241347150, 4.793316890450383, 5.482989704200729, 5.731102940099073, 6.480596833828254, 6.714204986471061, 7.552899562667763, 7.823308242891913, 8.390930751673256, 8.982002846163236, 9.244948603106116, 10.02758741544144, 10.28263640533346, 10.71616089464343, 11.42403267706662, 11.88064543471111, 12.11152099088996, 12.64660971198113, 13.22216015109653

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