Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 17 $
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 − 2·13-s − 2·15-s − 17-s + 4·19-s − 25-s + 27-s + 10·29-s − 8·31-s + 4·33-s + 2·37-s − 2·39-s − 10·41-s − 12·43-s − 2·45-s − 51-s − 6·53-s − 8·55-s + 4·57-s + 12·59-s − 10·61-s + 4·65-s + 12·67-s − 10·73-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 − 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.696·33-s + 0.328·37-s − 0.320·39-s − 1.56·41-s − 1.82·43-s − 0.298·45-s − 0.140·51-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 − 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(159936\)    =    \(2^{6} \cdot 3 \cdot 7^{2} \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{159936} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 159936,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.54672689325630, −13.09179148555876, −12.48044408050806, −11.99135271880035, −11.66420880266146, −11.39696779179956, −10.62431322042630, −10.00187421142739, −9.772079485155243, −9.109889051541514, −8.656800581549635, −8.277300400451674, −7.702789927065238, −7.245137953029305, −6.746684734057688, −6.393990573287284, −5.548059242155008, −4.918196818202853, −4.510755927228541, −3.886316755749253, −3.353430447496017, −3.084922600176579, −2.124228643432955, −1.596715652747368, −0.8523915268642746, 0, 0.8523915268642746, 1.596715652747368, 2.124228643432955, 3.084922600176579, 3.353430447496017, 3.886316755749253, 4.510755927228541, 4.918196818202853, 5.548059242155008, 6.393990573287284, 6.746684734057688, 7.245137953029305, 7.702789927065238, 8.277300400451674, 8.656800581549635, 9.109889051541514, 9.772079485155243, 10.00187421142739, 10.62431322042630, 11.39696779179956, 11.66420880266146, 11.99135271880035, 12.48044408050806, 13.09179148555876, 13.54672689325630

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