Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 167 $
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 + 4·7-s + 9-s − 4·11-s − 2·15-s − 4·17-s − 4·19-s + 4·21-s + 4·23-s − 25-s + 27-s + 2·29-s − 4·31-s − 4·33-s − 8·35-s + 12·37-s + 12·41-s − 8·43-s − 2·45-s + 9·49-s − 4·51-s − 14·53-s + 8·55-s − 4·57-s + 2·59-s + 2·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s − 0.970·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.35·35-s + 1.97·37-s + 1.87·41-s − 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.560·51-s − 1.92·53-s + 1.07·55-s − 0.529·57-s + 0.260·59-s + 0.256·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(32064\)    =    \(2^{6} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{32064} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 32064,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
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

−15.15455638703463, −14.89370982006500, −14.43579333999096, −13.77917505160416, −13.09025074569056, −12.86773148602701, −12.15963167540906, −11.36244442461163, −11.09358174603300, −10.78467399404691, −9.992351225555144, −9.219411247374466, −8.718094467428986, −8.110412932883341, −7.788309923916361, −7.465179944708445, −6.588831373369671, −5.888489552069625, −4.951034504195617, −4.649065869955131, −4.139454540996347, −3.316187585824588, −2.469851734177173, −2.060873104949273, −1.051956517441676, 0, 1.051956517441676, 2.060873104949273, 2.469851734177173, 3.316187585824588, 4.139454540996347, 4.649065869955131, 4.951034504195617, 5.888489552069625, 6.588831373369671, 7.465179944708445, 7.788309923916361, 8.110412932883341, 8.718094467428986, 9.219411247374466, 9.992351225555144, 10.78467399404691, 11.09358174603300, 11.36244442461163, 12.15963167540906, 12.86773148602701, 13.09025074569056, 13.77917505160416, 14.43579333999096, 14.89370982006500, 15.15455638703463

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