Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 2·5-s + 6-s + 3·8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s − 4·22-s − 3·24-s − 25-s − 2·26-s − 27-s − 2·29-s + 2·30-s − 5·32-s − 4·33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.883·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(147\)    =    \(3 \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{147} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 147,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7222862454$
$L(\frac12)$  $\approx$  $0.7222862454$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
7 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
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 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.11390515603707, −18.48505770001858, −17.60070085035207, −16.95975362441167, −16.44373210984857, −14.84576993456595, −13.95408359718786, −13.09697998821176, −11.98659319295798, −10.77789244508403, −9.863214598051417, −9.163045893798789, −7.977686193886307, −6.534849214747906, −5.485107655900711, −4.016718632194991, −1.407867712777502, 1.407867712777502, 4.016718632194991, 5.485107655900711, 6.534849214747906, 7.977686193886307, 9.163045893798789, 9.863214598051417, 10.77789244508403, 11.98659319295798, 13.09697998821176, 13.95408359718786, 14.84576993456595, 16.44373210984857, 16.95975362441167, 17.60070085035207, 18.48505770001858, 19.11390515603707

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