Properties

Degree $2$
Conductor $272832$
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 − 2·5-s + 9-s − 4·11-s − 2·13-s + 2·15-s − 2·17-s − 4·19-s − 25-s − 27-s − 29-s + 8·31-s + 4·33-s + 10·37-s + 2·39-s + 6·41-s − 12·43-s − 2·45-s + 8·47-s + 2·51-s − 6·53-s + 8·55-s + 4·57-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 1.43·31-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 1.82·43-s − 0.298·45-s + 1.16·47-s + 0.280·51-s − 0.824·53-s + 1.07·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272832\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 29\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{272832} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 272832,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9598787249\)
\(L(\frac12)\) \(\approx\) \(0.9598787249\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
29 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.75376726101126, −12.34870886186942, −11.63086899047820, −11.54151201338193, −11.04839698205534, −10.39304781481673, −10.19916251818318, −9.686337520561567, −9.005781728882542, −8.489238744604393, −8.024587978005645, −7.618880050431598, −7.275805343925423, −6.576150583032106, −6.158909186235998, −5.685851638489171, −5.000177695237483, −4.504441380779510, −4.354299451744864, −3.557652830907396, −2.944674437594282, −2.372035924014351, −1.886882030196808, −0.8011785468840494, −0.3656201895656679, 0.3656201895656679, 0.8011785468840494, 1.886882030196808, 2.372035924014351, 2.944674437594282, 3.557652830907396, 4.354299451744864, 4.504441380779510, 5.000177695237483, 5.685851638489171, 6.158909186235998, 6.576150583032106, 7.275805343925423, 7.618880050431598, 8.024587978005645, 8.489238744604393, 9.005781728882542, 9.686337520561567, 10.19916251818318, 10.39304781481673, 11.04839698205534, 11.54151201338193, 11.63086899047820, 12.34870886186942, 12.75376726101126

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