Properties

Degree 2
Conductor $ 3^{2} \cdot 19 $
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  + 2·2-s + 2·4-s − 5-s + 3·7-s − 2·10-s + 3·11-s − 6·13-s + 6·14-s − 4·16-s − 3·17-s − 19-s − 2·20-s + 6·22-s − 4·23-s − 4·25-s − 12·26-s + 6·28-s + 10·29-s + 2·31-s − 8·32-s − 6·34-s − 3·35-s + 8·37-s − 2·38-s + 8·41-s − 43-s + 6·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s − 0.632·10-s + 0.904·11-s − 1.66·13-s + 1.60·14-s − 16-s − 0.727·17-s − 0.229·19-s − 0.447·20-s + 1.27·22-s − 0.834·23-s − 4/5·25-s − 2.35·26-s + 1.13·28-s + 1.85·29-s + 0.359·31-s − 1.41·32-s − 1.02·34-s − 0.507·35-s + 1.31·37-s − 0.324·38-s + 1.24·41-s − 0.152·43-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(171\)    =    \(3^{2} \cdot 19\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{171} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 171,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.140174037$
$L(\frac12)$  $\approx$  $2.140174037$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;19\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;19\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
19 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−19.93432237990596, −19.51344019620567, −17.93095264578293, −17.39598276100219, −16.10736052345169, −15.12639059677174, −14.49831912406803, −13.94685934671170, −12.70402288938110, −11.84438281704763, −11.42285722847365, −9.886397115513209, −8.513296718878561, −7.331798691613333, −6.128319234976364, −4.752276349221271, −4.208834254933773, −2.440310117334424, 2.440310117334424, 4.208834254933773, 4.752276349221271, 6.128319234976364, 7.331798691613333, 8.513296718878561, 9.886397115513209, 11.42285722847365, 11.84438281704763, 12.70402288938110, 13.94685934671170, 14.49831912406803, 15.12639059677174, 16.10736052345169, 17.39598276100219, 17.93095264578293, 19.51344019620567, 19.93432237990596

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