Properties

Degree $2$
Conductor $235200$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·11-s − 2·13-s + 6·17-s − 4·19-s + 8·23-s − 27-s − 8·31-s + 6·33-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s − 4·47-s − 6·51-s − 6·53-s + 4·57-s + 6·59-s − 6·61-s − 8·69-s + 4·71-s − 12·73-s + 8·79-s + 81-s − 12·83-s − 14·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s − 0.192·27-s − 1.43·31-s + 1.04·33-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.583·47-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 0.781·59-s − 0.768·61-s − 0.963·69-s + 0.474·71-s − 1.40·73-s + 0.900·79-s + 1/9·81-s − 1.31·83-s − 1.48·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235200\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{235200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8681355318\)
\(L(\frac12)\) \(\approx\) \(0.8681355318\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.78031440348474, −12.55041134940184, −12.15984787342749, −11.35395747879404, −10.93835462233813, −10.75232910055680, −10.17846840766099, −9.726154285020004, −9.313871408396260, −8.588205828106489, −8.183695716924415, −7.584722424020981, −7.258065244375336, −6.887910136148184, −6.005807308249007, −5.617493697622276, −5.309841211131815, −4.725022351868239, −4.339592666905075, −3.408558324885929, −3.045271358240637, −2.443679161297821, −1.799639465211878, −1.037077695035889, −0.2944524137724034, 0.2944524137724034, 1.037077695035889, 1.799639465211878, 2.443679161297821, 3.045271358240637, 3.408558324885929, 4.339592666905075, 4.725022351868239, 5.309841211131815, 5.617493697622276, 6.005807308249007, 6.887910136148184, 7.258065244375336, 7.584722424020981, 8.183695716924415, 8.588205828106489, 9.313871408396260, 9.726154285020004, 10.17846840766099, 10.75232910055680, 10.93835462233813, 11.35395747879404, 12.15984787342749, 12.55041134940184, 12.78031440348474

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