Properties

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

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: \(2\)
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 \( 1 - 2 T + p T^{2} \)
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

−13.63633022190879, −13.22923491757054, −12.69497597391282, −12.12473486327329, −11.83878525284584, −11.39024349450144, −10.82260408085068, −10.27617426586131, −9.860996062951584, −9.466915056582372, −8.801581429105702, −8.446719585784534, −7.971851600230843, −7.289249274185941, −6.901159527969067, −6.398469324932067, −5.868724074222735, −5.372792006762718, −4.637189617185135, −4.256782210824661, −3.795843598590341, −2.974531916976138, −2.565940748683696, −1.648387417074006, −1.472034785642845, 0, 0, 1.472034785642845, 1.648387417074006, 2.565940748683696, 2.974531916976138, 3.795843598590341, 4.256782210824661, 4.637189617185135, 5.372792006762718, 5.868724074222735, 6.398469324932067, 6.901159527969067, 7.289249274185941, 7.971851600230843, 8.446719585784534, 8.801581429105702, 9.466915056582372, 9.860996062951584, 10.27617426586131, 10.82260408085068, 11.39024349450144, 11.83878525284584, 12.12473486327329, 12.69497597391282, 13.22923491757054, 13.63633022190879

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