Properties

Degree $2$
Conductor $27200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 4·7-s + 9-s − 6·11-s + 2·13-s + 17-s + 4·19-s − 8·21-s + 4·27-s − 4·31-s + 12·33-s − 4·37-s − 4·39-s + 6·41-s + 8·43-s + 9·49-s − 2·51-s − 6·53-s − 8·57-s + 4·61-s + 4·63-s + 8·67-s − 2·73-s − 24·77-s + 8·79-s − 11·81-s − 6·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 0.242·17-s + 0.917·19-s − 1.74·21-s + 0.769·27-s − 0.718·31-s + 2.08·33-s − 0.657·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 1.05·57-s + 0.512·61-s + 0.503·63-s + 0.977·67-s − 0.234·73-s − 2.73·77-s + 0.900·79-s − 1.22·81-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27200\)    =    \(2^{6} \cdot 5^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{27200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 27200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.421653570\)
\(L(\frac12)\) \(\approx\) \(1.421653570\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−15.36672924692005, −14.77434425034852, −13.99834635066318, −13.89214489840302, −12.93270526742218, −12.55353300033688, −11.97207921920784, −11.26850598866836, −11.04097667132281, −10.68448108968749, −10.04072546969267, −9.312954974002456, −8.453950517255844, −8.100031990709498, −7.491070150159299, −7.035739237199059, −5.943570281995381, −5.667205781469574, −5.052646513213127, −4.789859151280367, −3.873934299723278, −2.942029838858224, −2.202686268899869, −1.315874239607975, −0.5375578424519806, 0.5375578424519806, 1.315874239607975, 2.202686268899869, 2.942029838858224, 3.873934299723278, 4.789859151280367, 5.052646513213127, 5.667205781469574, 5.943570281995381, 7.035739237199059, 7.491070150159299, 8.100031990709498, 8.453950517255844, 9.312954974002456, 10.04072546969267, 10.68448108968749, 11.04097667132281, 11.26850598866836, 11.97207921920784, 12.55353300033688, 12.93270526742218, 13.89214489840302, 13.99834635066318, 14.77434425034852, 15.36672924692005

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