Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 31 $
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 + 3·5-s − 6-s − 2·7-s − 8-s + 9-s − 3·10-s + 5·11-s + 12-s − 7·13-s + 2·14-s + 3·15-s + 16-s − 17-s − 18-s + 7·19-s + 3·20-s − 2·21-s − 5·22-s + 4·23-s − 24-s + 4·25-s + 7·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.948·10-s + 1.50·11-s + 0.288·12-s − 1.94·13-s + 0.534·14-s + 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.60·19-s + 0.670·20-s − 0.436·21-s − 1.06·22-s + 0.834·23-s − 0.204·24-s + 4/5·25-s + 1.37·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 186 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(186\)    =    \(2 \cdot 3 \cdot 31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{186} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 186,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.182357297$
$L(\frac12)$  $\approx$  $1.182357297$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;31\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;31\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
31 \( 1 + T \)
good5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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

−19.74945369374624, −18.88604760887138, −17.93254907428713, −17.00024915577686, −16.70848558656099, −15.25425303131305, −14.44426610018239, −13.65787696802804, −12.60131264603828, −11.61273048285036, −10.05252486013069, −9.538738513600828, −9.061010670239073, −7.362608270618632, −6.624757501665547, −5.255665283675663, −3.219350549310240, −1.823389024561013, 1.823389024561013, 3.219350549310240, 5.255665283675663, 6.624757501665547, 7.362608270618632, 9.061010670239073, 9.538738513600828, 10.05252486013069, 11.61273048285036, 12.60131264603828, 13.65787696802804, 14.44426610018239, 15.25425303131305, 16.70848558656099, 17.00024915577686, 17.93254907428713, 18.88604760887138, 19.74945369374624

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