Properties

Degree 2
Conductor $ 2^{6} \cdot 3 $
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 + 2·13-s + 2·15-s − 6·17-s + 4·19-s + 4·21-s − 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 8·35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 2·45-s + 8·47-s + 9·49-s + 6·51-s − 10·53-s + 8·55-s − 4·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-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 + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s − 1.37·53-s + 1.07·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(192\)    =    \(2^{6} \cdot 3\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{192} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 192,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.93453071153368, −19.12476033647996, −18.41991304342513, −17.51400173212347, −16.23743873243261, −15.87107887160036, −15.30663303021776, −13.54286180230792, −13.02873657261512, −12.03546217958315, −11.10886283085596, −10.20586918892849, −9.133901360741508, −7.831823947573225, −6.809468979037549, −5.770924738437827, −4.311587892181168, −3.005711630917525, 0, 3.005711630917525, 4.311587892181168, 5.770924738437827, 6.809468979037549, 7.831823947573225, 9.133901360741508, 10.20586918892849, 11.10886283085596, 12.03546217958315, 13.02873657261512, 13.54286180230792, 15.30663303021776, 15.87107887160036, 16.23743873243261, 17.51400173212347, 18.41991304342513, 19.12476033647996, 19.93453071153368

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