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 − 2·13-s + 2·15-s + 6·17-s − 4·19-s − 21-s − 4·23-s − 25-s + 27-s + 6·29-s − 8·31-s − 2·35-s − 10·37-s − 2·39-s − 10·41-s + 12·43-s + 2·45-s − 8·47-s + 49-s + 6·51-s + 6·53-s − 4·57-s + 4·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.338·35-s − 1.64·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-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  =  $(2,\ 168,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.451262370$
$L(\frac12)$  $\approx$  $1.451262370$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.87014417510300, −19.17240317065542, −18.28307608141354, −17.33414154732982, −16.54592794024899, −15.51211073526372, −14.42429832914956, −13.91550367121578, −12.82992732310358, −12.06652674974972, −10.42205409739276, −9.848554569012740, −8.820314983809013, −7.659946210194031, −6.421366582935158, −5.250192136389902, −3.569214080812381, −2.065156929453541, 2.065156929453541, 3.569214080812381, 5.250192136389902, 6.421366582935158, 7.659946210194031, 8.820314983809013, 9.848554569012740, 10.42205409739276, 12.06652674974972, 12.82992732310358, 13.91550367121578, 14.42429832914956, 15.51211073526372, 16.54592794024899, 17.33414154732982, 18.28307608141354, 19.17240317065542, 19.87014417510300

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