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  + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 8·23-s − 6·29-s − 8·31-s + 2·37-s + 2·41-s − 12·43-s − 8·47-s + 6·53-s − 4·59-s + 2·61-s + 12·67-s + 8·71-s − 14·73-s + 12·83-s + 2·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 1.82·43-s − 1.16·47-s + 0.824·53-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 1.63·73-s + 1.31·83-s + 0.211·89-s + 1.01·97-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

\(L(1)\) \(\approx\) \(1.706254685\)
\(L(\frac12)\) \(\approx\) \(1.706254685\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.22250772499897, −12.80355363496028, −12.25337633593195, −11.68392107821497, −11.30706053408288, −10.91901301809615, −10.41920844745713, −9.700196971764083, −9.378302567329046, −8.916463899286581, −8.534834111210672, −7.887755010234144, −7.253600138216777, −6.861924704243304, −6.511949136533669, −5.875815426505951, −5.231649842502073, −4.802333000208322, −4.240031017453571, −3.569918717847688, −3.285199667348784, −2.327984139989925, −1.898271404986577, −1.239706676228107, −0.3808751481519590, 0.3808751481519590, 1.239706676228107, 1.898271404986577, 2.327984139989925, 3.285199667348784, 3.569918717847688, 4.240031017453571, 4.802333000208322, 5.231649842502073, 5.875815426505951, 6.511949136533669, 6.861924704243304, 7.253600138216777, 7.887755010234144, 8.534834111210672, 8.916463899286581, 9.378302567329046, 9.700196971764083, 10.41920844745713, 10.91901301809615, 11.30706053408288, 11.68392107821497, 12.25337633593195, 12.80355363496028, 13.22250772499897

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