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 − 6·11-s + 13-s + 6·17-s + 5·19-s − 8·23-s − 27-s + 2·29-s − 4·31-s + 6·33-s + 37-s − 39-s − 6·41-s − 12·43-s − 12·47-s − 6·51-s + 6·53-s − 5·57-s + 8·59-s − 15·61-s − 5·67-s + 8·69-s − 6·71-s + 73-s − 79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 1.45·17-s + 1.14·19-s − 1.66·23-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 1.04·33-s + 0.164·37-s − 0.160·39-s − 0.937·41-s − 1.82·43-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.662·57-s + 1.04·59-s − 1.92·61-s − 0.610·67-s + 0.963·69-s − 0.712·71-s + 0.117·73-s − 0.112·79-s + 1/9·81-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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−13.29041741693713, −12.53181319168236, −12.24073709023576, −11.69346560676811, −11.45076986867105, −10.74570456356884, −10.18016876342700, −10.06279642858307, −9.727169143625372, −8.869033061908792, −8.264800011861084, −7.955460403417664, −7.485701908450039, −7.124407647657453, −6.262389622456099, −5.950537946568107, −5.385602648618705, −5.049597363675438, −4.628598726736305, −3.695673402834303, −3.310590889873801, −2.848960120575428, −1.939788011901327, −1.549157132262993, −0.6356746144320483, 0, 0.6356746144320483, 1.549157132262993, 1.939788011901327, 2.848960120575428, 3.310590889873801, 3.695673402834303, 4.628598726736305, 5.049597363675438, 5.385602648618705, 5.950537946568107, 6.262389622456099, 7.124407647657453, 7.485701908450039, 7.955460403417664, 8.264800011861084, 8.869033061908792, 9.727169143625372, 10.06279642858307, 10.18016876342700, 10.74570456356884, 11.45076986867105, 11.69346560676811, 12.24073709023576, 12.53181319168236, 13.29041741693713

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