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 + 8·19-s − 6·29-s − 4·31-s + 10·37-s − 6·41-s − 4·43-s − 6·53-s + 12·59-s + 10·61-s − 4·67-s + 12·71-s − 10·73-s − 8·79-s + 12·83-s − 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.554·13-s + 1.45·17-s + 1.83·19-s − 1.11·29-s − 0.718·31-s + 1.64·37-s − 0.937·41-s − 0.609·43-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s + 1.31·83-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: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.156920464\)
\(L(\frac12)\) \(\approx\) \(3.156920464\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.30912603691762, −12.68647625537679, −12.26060063528297, −11.65836689792462, −11.35650661599666, −10.99639118652184, −10.12882293079986, −9.884167646086987, −9.498184703293588, −8.935614631813923, −8.299716026603124, −7.881358410738993, −7.406840957496248, −7.018338932395977, −6.298777926797488, −5.737854248383668, −5.348252439510580, −4.981206914612833, −4.056353987023267, −3.621639900926514, −3.188395207066793, −2.575525016394423, −1.700758950809128, −1.201406989531250, −0.5556913153217068, 0.5556913153217068, 1.201406989531250, 1.700758950809128, 2.575525016394423, 3.188395207066793, 3.621639900926514, 4.056353987023267, 4.981206914612833, 5.348252439510580, 5.737854248383668, 6.298777926797488, 7.018338932395977, 7.406840957496248, 7.881358410738993, 8.299716026603124, 8.935614631813923, 9.498184703293588, 9.884167646086987, 10.12882293079986, 10.99639118652184, 11.35650661599666, 11.65836689792462, 12.26060063528297, 12.68647625537679, 13.30912603691762

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