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

−14.97354348298255, −14.32524200286004, −13.86756144104277, −13.59381691833333, −12.84630744858541, −12.68077855410029, −11.91108680160799, −11.28551608154144, −10.59821457899757, −10.23369533268342, −9.703637746912206, −9.202256631420371, −8.752258441185727, −7.947733458070073, −7.515819141452314, −7.147123509279834, −6.200443132405007, −5.664507676254207, −5.276368609168612, −4.549768021220500, −3.822537174132643, −2.997814565122510, −2.593854119500655, −1.897176430948922, −1.161356443259139, 0, 1.161356443259139, 1.897176430948922, 2.593854119500655, 2.997814565122510, 3.822537174132643, 4.549768021220500, 5.276368609168612, 5.664507676254207, 6.200443132405007, 7.147123509279834, 7.515819141452314, 7.947733458070073, 8.752258441185727, 9.202256631420371, 9.703637746912206, 10.23369533268342, 10.59821457899757, 11.28551608154144, 11.91108680160799, 12.68077855410029, 12.84630744858541, 13.59381691833333, 13.86756144104277, 14.32524200286004, 14.97354348298255

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