Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·13-s + 2·19-s − 6·29-s + 8·31-s + 4·37-s + 6·41-s + 2·43-s − 6·47-s + 6·53-s − 12·59-s − 8·61-s + 2·67-s + 6·71-s + 2·73-s + 16·79-s + 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.554·13-s + 0.458·19-s − 1.11·29-s + 1.43·31-s + 0.657·37-s + 0.937·41-s + 0.304·43-s − 0.875·47-s + 0.824·53-s − 1.56·59-s − 1.02·61-s + 0.244·67-s + 0.712·71-s + 0.234·73-s + 1.80·79-s + 0.635·89-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: \(1\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.49578112448531, −12.94088982829378, −12.48002584536418, −12.02823824686007, −11.47202202276700, −11.08780491198522, −10.66186746162246, −10.08325494998839, −9.554491551474145, −9.203145114632914, −8.686361383010448, −8.006856129215390, −7.753641133008332, −7.210395703236182, −6.475459964170059, −6.171920924210468, −5.664534271363509, −4.949553274381565, −4.604280874234235, −3.825077052503235, −3.498599217437739, −2.718719589028006, −2.270954061144040, −1.400638559758660, −0.9208817541614280, 0, 0.9208817541614280, 1.400638559758660, 2.270954061144040, 2.718719589028006, 3.498599217437739, 3.825077052503235, 4.604280874234235, 4.949553274381565, 5.664534271363509, 6.171920924210468, 6.475459964170059, 7.210395703236182, 7.753641133008332, 8.006856129215390, 8.686361383010448, 9.203145114632914, 9.554491551474145, 10.08325494998839, 10.66186746162246, 11.08780491198522, 11.47202202276700, 12.02823824686007, 12.48002584536418, 12.94088982829378, 13.49578112448531

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