Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 2·11-s − 2·13-s − 4·17-s + 8·23-s − 27-s − 2·31-s − 2·33-s + 8·37-s + 2·39-s + 2·41-s + 2·43-s − 10·47-s + 4·51-s − 2·53-s − 4·59-s − 10·61-s − 2·67-s − 8·69-s + 12·71-s + 10·73-s − 16·79-s + 81-s + 16·83-s − 14·89-s + 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 0.970·17-s + 1.66·23-s − 0.192·27-s − 0.359·31-s − 0.348·33-s + 1.31·37-s + 0.320·39-s + 0.312·41-s + 0.304·43-s − 1.45·47-s + 0.560·51-s − 0.274·53-s − 0.520·59-s − 1.28·61-s − 0.244·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s − 1.80·79-s + 1/9·81-s + 1.75·83-s − 1.48·89-s + 0.207·93-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: \(1\)
Selberg data: \((2,\ 235200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
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

−13.16256979560643, −12.60223923111472, −12.30577818081780, −11.61959074303553, −11.26829438308090, −10.94367730245709, −10.52022472206954, −9.708584512082795, −9.524054231642834, −8.994292594486510, −8.560757958609402, −7.786824313900745, −7.509451973311216, −6.721797759069944, −6.633818315105028, −6.028100765531776, −5.402949843087116, −4.864712870606050, −4.527207162771662, −3.970945599727420, −3.251640623447076, −2.737826859896236, −2.061472297585932, −1.395555060366562, −0.7667401732532654, 0, 0.7667401732532654, 1.395555060366562, 2.061472297585932, 2.737826859896236, 3.251640623447076, 3.970945599727420, 4.527207162771662, 4.864712870606050, 5.402949843087116, 6.028100765531776, 6.633818315105028, 6.721797759069944, 7.509451973311216, 7.786824313900745, 8.560757958609402, 8.994292594486510, 9.524054231642834, 9.708584512082795, 10.52022472206954, 10.94367730245709, 11.26829438308090, 11.61959074303553, 12.30577818081780, 12.60223923111472, 13.16256979560643

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