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  + 2·13-s + 6·17-s − 4·19-s + 6·29-s − 4·31-s − 2·37-s + 6·41-s + 8·43-s − 12·47-s + 6·53-s + 12·59-s − 2·61-s + 8·67-s + 14·73-s + 16·79-s + 12·83-s + 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 1.75·47-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.977·67-s + 1.63·73-s + 1.80·79-s + 1.31·83-s + 0.635·89-s + 1.42·97-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

\(L(1)\) \(\approx\) \(3.422630537\)
\(L(\frac12)\) \(\approx\) \(3.422630537\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.04650523793161, −12.68223974161961, −12.36678256386288, −11.73337585803760, −11.34192117467100, −10.71921527549252, −10.43645320728373, −9.863144958068950, −9.423539838474794, −8.830525050169621, −8.402644855485250, −7.871615258004236, −7.522634809403516, −6.799729495598001, −6.321078953820848, −5.956162002950151, −5.187702186839038, −4.957899751289151, −4.096950467472133, −3.658124828265495, −3.215411802430124, −2.373431367680353, −1.961542205898623, −1.011027920890447, −0.6351287585130339, 0.6351287585130339, 1.011027920890447, 1.961542205898623, 2.373431367680353, 3.215411802430124, 3.658124828265495, 4.096950467472133, 4.957899751289151, 5.187702186839038, 5.956162002950151, 6.321078953820848, 6.799729495598001, 7.522634809403516, 7.871615258004236, 8.402644855485250, 8.830525050169621, 9.423539838474794, 9.863144958068950, 10.43645320728373, 10.71921527549252, 11.34192117467100, 11.73337585803760, 12.36678256386288, 12.68223974161961, 13.04650523793161

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