Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 131 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2·5-s − 6-s − 2·7-s − 8-s + 9-s + 2·10-s + 3·11-s + 12-s − 5·13-s + 2·14-s − 2·15-s + 16-s + 7·17-s − 18-s − 5·19-s − 2·20-s − 2·21-s − 3·22-s − 4·23-s − 24-s − 25-s + 5·26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.904·11-s + 0.288·12-s − 1.38·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 1.14·19-s − 0.447·20-s − 0.436·21-s − 0.639·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.980·26-s + 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(786\)    =    \(2 \cdot 3 \cdot 131\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{786} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 786,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;131\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;131\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
131 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−19.44917725820080, −19.16957778086373, −18.58487438617924, −17.36392096980254, −16.78040585557907, −16.22410504657240, −15.28506831294982, −14.75371568315357, −14.07982247764675, −12.80593233332864, −12.16739160855351, −11.65873422960627, −10.32790526177037, −9.864701877333940, −9.007660079477580, −8.184065983773168, −7.391216058833113, −6.776860191708650, −5.500042069158619, −3.996546450021363, −3.304718306795995, −1.908138085338151, 0, 1.908138085338151, 3.304718306795995, 3.996546450021363, 5.500042069158619, 6.776860191708650, 7.391216058833113, 8.184065983773168, 9.007660079477580, 9.864701877333940, 10.32790526177037, 11.65873422960627, 12.16739160855351, 12.80593233332864, 14.07982247764675, 14.75371568315357, 15.28506831294982, 16.22410504657240, 16.78040585557907, 17.36392096980254, 18.58487438617924, 19.16957778086373, 19.44917725820080

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