Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 7-s + 9-s + 6·13-s − 2·15-s − 2·17-s + 4·19-s − 21-s − 4·23-s − 25-s − 27-s − 10·29-s − 8·31-s + 2·35-s + 6·37-s − 6·39-s − 2·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 2·51-s − 10·53-s − 4·57-s + 12·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.338·35-s + 0.986·37-s − 0.960·39-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{168} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 168,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.160754652$
$L(\frac12)$  $\approx$  $1.160754652$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
7 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−18.70394023364009, −18.16576263279759, −17.49940938333795, −16.48464094826355, −15.77292720845454, −14.58166642231685, −13.56609089328758, −12.98797626118105, −11.57282372259661, −10.95625562106234, −9.826688574335115, −8.859066757414731, −7.515634816193186, −6.140744237077391, −5.471375350456022, −3.852825531048511, −1.720659913548846, 1.720659913548846, 3.852825531048511, 5.471375350456022, 6.140744237077391, 7.515634816193186, 8.859066757414731, 9.826688574335115, 10.95625562106234, 11.57282372259661, 12.98797626118105, 13.56609089328758, 14.58166642231685, 15.77292720845454, 16.48464094826355, 17.49940938333795, 18.16576263279759, 18.70394023364009

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