↑

Properties

Label 2-882-63.4-c1-0-22
Degree $2$
Conductor $882$
Sign $0.995 - 0.0931i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
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 + (1.09 + 1.34i)3-s + (−0.499 − 0.866i)4-s + 3.18·5-s + (1.71 − 0.272i)6-s − 0.999·8-s + (−0.619 + 2.93i)9-s + (1.59 − 2.75i)10-s + 3.18·11-s + (0.619 − 1.61i)12-s + (−2.85 + 4.93i)13-s + (3.47 + 4.28i)15-s + (−0.5 + 0.866i)16-s + (0.760 − 1.31i)17-s + (2.23 + 2.00i)18-s + (0.641 + 1.11i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.629 + 0.776i)3-s + (−0.249 − 0.433i)4-s + 1.42·5-s + (0.698 − 0.111i)6-s − 0.353·8-s + (−0.206 + 0.978i)9-s + (0.503 − 0.871i)10-s + 0.959·11-s + (0.178 − 0.466i)12-s + (−0.790 + 1.36i)13-s + (0.896 + 1.10i)15-s + (−0.125 + 0.216i)16-s + (0.184 − 0.319i)17-s + (0.526 + 0.472i)18-s + (0.147 + 0.254i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.995 - 0.0931i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.995 - 0.0931i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84654 + 0.132839i\)
\(L(\frac12)\) \(\approx\) \(2.84654 + 0.132839i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 1.34i)T \)
7 \( 1 \)
good5 \( 1 - 3.18T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (2.85 - 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.760 + 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.641 - 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.23T + 23T^{2} \)
29 \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 + 8.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.80 + 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.562 + 0.974i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (-2.48 + 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.06 + 3.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 - 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.112 + 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.42 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.812290355918444179311839590916, −9.471039271977516877638993739137, −9.113672828888600354905626149545, −7.68455063741365624215902771319, −6.46852038744802868379269643520, −5.59959806119460646535982743994, −4.62514239265376033564564081336, −3.81064746550542420797907432895, −2.48828204563254151578096868014, −1.78828698966792937214516640034, 1.35123332733450804489928228884, 2.60515980569640017113562494682, 3.57886166778538079775157228155, 5.18993893585747778876285429393, 5.80074045747594573703804900498, 6.78029053849040056732479832092, 7.32977581941157334858324290660, 8.452838576630676083404162524219, 9.146392080122039511627017178263, 9.832032870424377222068017624417

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