Properties

Degree $2$
Conductor $42432$
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 + 2·5-s + 9-s + 4·11-s − 13-s − 2·15-s + 17-s − 4·19-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s + 2·37-s + 39-s + 2·41-s − 4·43-s + 2·45-s − 8·47-s − 7·49-s − 51-s + 10·53-s + 8·55-s + 4·57-s + 4·59-s − 14·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s + 0.242·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s − 49-s − 0.140·51-s + 1.37·53-s + 1.07·55-s + 0.529·57-s + 0.520·59-s − 1.79·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(42432\)    =    \(2^{6} \cdot 3 \cdot 13 \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{42432} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 42432,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
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

−14.89984566370855, −14.45122345536960, −13.95611374465542, −13.31115334946585, −13.06269933508733, −12.15524335750224, −11.98804628422023, −11.39904016741268, −10.76019642005738, −10.19434036060814, −9.769319480593601, −9.349638128896503, −8.620865664689243, −8.172196168122655, −7.336073110140607, −6.736980275684719, −6.202592633187998, −5.989995664435743, −5.139188178520754, −4.551654048702153, −4.057151492879122, −3.185389582115792, −2.434810127094690, −1.651673467737119, −1.121387818773928, 0, 1.121387818773928, 1.651673467737119, 2.434810127094690, 3.185389582115792, 4.057151492879122, 4.551654048702153, 5.139188178520754, 5.989995664435743, 6.202592633187998, 6.736980275684719, 7.336073110140607, 8.172196168122655, 8.620865664689243, 9.349638128896503, 9.769319480593601, 10.19434036060814, 10.76019642005738, 11.39904016741268, 11.98804628422023, 12.15524335750224, 13.06269933508733, 13.31115334946585, 13.95611374465542, 14.45122345536960, 14.89984566370855

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