↑

Properties

Label 2-882-63.47-c1-0-26
Degree $2$
Conductor $882$
Sign $-0.300 + 0.953i$
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  + (0.866 − 0.5i)2-s + (0.284 + 1.70i)3-s + (0.499 − 0.866i)4-s − 3.89·5-s + (1.10 + 1.33i)6-s − 0.999i·8-s + (−2.83 + 0.971i)9-s + (−3.36 + 1.94i)10-s − 3.94i·11-s + (1.62 + 0.608i)12-s + (2.46 − 1.42i)13-s + (−1.10 − 6.64i)15-s + (−0.5 − 0.866i)16-s + (−0.371 − 0.642i)17-s + (−1.97 + 2.26i)18-s + (−1.54 − 0.892i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.164 + 0.986i)3-s + (0.249 − 0.433i)4-s − 1.74·5-s + (0.449 + 0.546i)6-s − 0.353i·8-s + (−0.946 + 0.323i)9-s + (−1.06 + 0.615i)10-s − 1.18i·11-s + (0.468 + 0.175i)12-s + (0.684 − 0.395i)13-s + (−0.285 − 1.71i)15-s + (−0.125 − 0.216i)16-s + (−0.0899 − 0.155i)17-s + (−0.464 + 0.532i)18-s + (−0.354 − 0.204i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.300 + 0.953i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.300 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571467 - 0.779006i\)
\(L(\frac12)\) \(\approx\) \(0.571467 - 0.779006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.284 - 1.70i)T \)
7 \( 1 \)
good5 \( 1 + 3.89T + 5T^{2} \)
11 \( 1 + 3.94iT - 11T^{2} \)
13 \( 1 + (-2.46 + 1.42i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.371 + 0.642i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.54 + 0.892i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.25iT - 23T^{2} \)
29 \( 1 + (2.50 + 1.44i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.04 + 1.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.50 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.24 + 9.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.471 + 0.816i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 - 1.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0105 + 0.0183i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.13 + 1.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.72 - 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.94iT - 71T^{2} \)
73 \( 1 + (4.20 - 2.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.81 + 3.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 - 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \)
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

−10.24119193453179832940759500728, −8.755573646461992434506205836808, −8.513974155111308779572843032947, −7.42655505495050752405014157363, −6.20197125406377861593223763784, −5.19248137664634223877858959183, −4.18268571986788444079750236121, −3.66097896407340022964442814670, −2.82070576224607788109117132660, −0.36975928888650739037506355311, 1.69664591407174090970163416479, 3.25461499558802942468557148845, 3.99442828999576379589939364277, 5.02679610109762894029574495337, 6.28824466701005562995654651183, 7.16000255446968174556237488001, 7.60281667804674523960733732183, 8.345748284069598904426429187031, 9.215914958789422751555173895772, 10.75357778451606323676483599468

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