Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 31 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s + 3·17-s + 18-s − 5·19-s + 20-s − 2·21-s − 3·22-s + 4·23-s + 24-s − 4·25-s − 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.14·19-s + 0.223·20-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-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  =  $(2,\ 186,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.952897668$
$L(\frac12)$  $\approx$  $1.952897668$
$L(\frac{3}{2})$   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 \( 1 - T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−19.59084504662377, −19.15289780072279, −18.04400420808399, −16.91064444610399, −16.08097103009146, −15.15813979406405, −14.46390693481413, −13.25276020234328, −13.05040591723051, −11.86609793920404, −10.48645678616775, −9.801450505814554, −8.512584183321875, −7.333193673252954, −6.214498978411575, −5.036205164710277, −3.544631869626427, −2.345469371255781, 2.345469371255781, 3.544631869626427, 5.036205164710277, 6.214498978411575, 7.333193673252954, 8.512584183321875, 9.801450505814554, 10.48645678616775, 11.86609793920404, 13.05040591723051, 13.25276020234328, 14.46390693481413, 15.15813979406405, 16.08097103009146, 16.91064444610399, 18.04400420808399, 19.15289780072279, 19.59084504662377

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