↑

Properties

Label 2-882-63.20-c1-0-6
Degree $2$
Conductor $882$
Sign $-0.380 - 0.924i$
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 + (−1.43 − 0.974i)3-s + (0.499 + 0.866i)4-s + (1.80 + 3.13i)5-s + (−0.752 − 1.56i)6-s + 0.999i·8-s + (1.09 + 2.79i)9-s + 3.61i·10-s + (1.73 + 1.00i)11-s + (0.128 − 1.72i)12-s + (−2.95 + 1.70i)13-s + (0.465 − 6.25i)15-s + (−0.5 + 0.866i)16-s − 6.17·17-s + (−0.443 + 2.96i)18-s − 1.01i·19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.826 − 0.562i)3-s + (0.249 + 0.433i)4-s + (0.809 + 1.40i)5-s + (−0.307 − 0.636i)6-s + 0.353i·8-s + (0.366 + 0.930i)9-s + 1.14i·10-s + (0.523 + 0.302i)11-s + (0.0370 − 0.498i)12-s + (−0.818 + 0.472i)13-s + (0.120 − 1.61i)15-s + (−0.125 + 0.216i)16-s − 1.49·17-s + (−0.104 + 0.699i)18-s − 0.232i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.380 - 0.924i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.927027 + 1.38436i\)
\(L(\frac12)\) \(\approx\) \(0.927027 + 1.38436i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 0.974i)T \)
7 \( 1 \)
good5 \( 1 + (-1.80 - 3.13i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.73 - 1.00i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.95 - 1.70i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 + 1.01iT - 19T^{2} \)
23 \( 1 + (-2.62 + 1.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.04 - 2.91i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.787 - 0.454i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.33T + 37T^{2} \)
41 \( 1 + (-2.85 - 4.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.39 - 4.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.11 - 1.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.75iT - 53T^{2} \)
59 \( 1 + (-4.49 - 7.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.7 - 7.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 7.20i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.466iT - 71T^{2} \)
73 \( 1 + 4.21iT - 73T^{2} \)
79 \( 1 + (1.91 - 3.31i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.00 + 6.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.78T + 89T^{2} \)
97 \( 1 + (-10.1 - 5.87i)T + (48.5 + 84.0i)T^{2} \)
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

−10.57751045733297337845414752165, −9.798696678894867559059629135043, −8.617244276082340623668369846115, −7.19488071405138763785813960169, −6.82566158608493545443465218754, −6.34600139761686583724180822589, −5.25725095354072948809471964427, −4.36992601418983523471842980006, −2.81252123522069729099111091976, −1.95015782165380097366666035260, 0.70851202143179514066166979933, 2.09869144020704641469760301887, 3.75098772698276330874985606113, 4.76388454176543541952920592719, 5.19913988176945092250309426266, 6.07853671439369510104895434598, 6.93038622058270277643828907729, 8.538916191966004959357691931868, 9.247059390313089784074567353015, 9.929699104273912930654693215417

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