↑

Properties

Label 2-882-63.4-c1-0-31
Degree $2$
Conductor $882$
Sign $0.373 + 0.927i$
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 + (1.68 − 0.396i)3-s + (−0.499 − 0.866i)4-s + 1.37·5-s + (0.5 − 1.65i)6-s − 0.999·8-s + (2.68 − 1.33i)9-s + (0.686 − 1.18i)10-s + 4.37·11-s + (−1.18 − 1.26i)12-s + (1 − 1.73i)13-s + (2.31 − 0.543i)15-s + (−0.5 + 0.866i)16-s + (−2.18 + 3.78i)17-s + (0.186 − 2.99i)18-s + (2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.973 − 0.228i)3-s + (−0.249 − 0.433i)4-s + 0.613·5-s + (0.204 − 0.677i)6-s − 0.353·8-s + (0.895 − 0.445i)9-s + (0.216 − 0.375i)10-s + 1.31·11-s + (−0.342 − 0.364i)12-s + (0.277 − 0.480i)13-s + (0.597 − 0.140i)15-s + (−0.125 + 0.216i)16-s + (−0.530 + 0.918i)17-s + (0.0438 − 0.705i)18-s + (0.573 + 0.993i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.373 + 0.927i$
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: \((2,\ 882,\ (\ :1/2),\ 0.373 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46010 - 1.66191i\)
\(L(\frac12)\) \(\approx\) \(2.46010 - 1.66191i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.396i)T \)
7 \( 1 \)
good5 \( 1 - 1.37T + 5T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.37T + 23T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.18 + 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.37 - 2.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.55 + 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (2.55 - 4.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.74 + 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.62 - 2.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.55 - 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.986673387976032675423288837598, −9.212632112042383134348355725637, −8.501326700783915964702842922532, −7.54442407523677282138486433969, −6.34926936412669383505700528566, −5.71541173447312992321421994251, −4.06701900198906911448336074545, −3.68067271713760049751381038426, −2.23625729788867751514618077490, −1.45368049449275543607068222721, 1.73407927815907374175459666774, 2.99243127385315125048316031527, 4.07863961872967698723859359383, 4.82306960102525839619598381578, 6.14257766890731202948227007977, 6.81539802458336483448266940896, 7.74918462882129830111135915140, 8.653119239296464640888148963967, 9.518699288257019744151421761637, 9.699834940105487695726194294647

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