↑

Properties

Label 2-462-7.4-c1-0-6
Degree $2$
Conductor $462$
Sign $0.991 - 0.126i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s + (2 − 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + 4·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.499 − 0.866i)18-s + (1.5 + 2.59i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s + (0.755 − 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.121 − 0.210i)17-s + (0.117 − 0.204i)18-s + (0.344 + 0.596i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96994 + 0.125012i\)
\(L(\frac12)\) \(\approx\) \(1.96994 + 0.125012i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2 + 1.73i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 2.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 + (-4 - 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.18739641860029402026088233978, −10.22562872242445896207014746238, −8.876802409750981891691028317180, −8.233858461217780113791476774895, −7.38473853408885405857455393289, −6.50394873964644615370761367126, −5.48608931412836651163902807347, −4.28786695802210391199229024100, −3.22274983635293859585433773869, −1.36724753470873787478607688365, 1.66011252726681229380229248115, 3.00935733405640523267731201922, 4.13199292402678315299490993059, 5.13112054639530688209121993443, 6.03396418144715189455885145866, 7.49786838631527557145817345188, 8.679677599137892028760317313119, 9.175670119333603190562741416483, 10.28900000896782880103717838007, 11.17771582984605063912808312856

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