↑

Properties

Label 2-882-147.101-c1-0-13
Degree $2$
Conductor $882$
Sign $0.732 - 0.681i$
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  + (0.930 + 0.365i)2-s + (0.733 + 0.680i)4-s + (3.14 + 2.14i)5-s + (−0.923 − 2.47i)7-s + (0.433 + 0.900i)8-s + (2.14 + 3.14i)10-s + (0.0766 + 0.508i)11-s + (1.07 − 0.856i)13-s + (0.0461 − 2.64i)14-s + (0.0747 + 0.997i)16-s + (4.86 + 1.50i)17-s + (0.729 − 0.421i)19-s + (0.847 + 3.71i)20-s + (−0.114 + 0.501i)22-s + (−0.673 − 2.18i)23-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.366 + 0.340i)4-s + (1.40 + 0.959i)5-s + (−0.349 − 0.937i)7-s + (0.153 + 0.318i)8-s + (0.678 + 0.995i)10-s + (0.0231 + 0.153i)11-s + (0.297 − 0.237i)13-s + (0.0123 − 0.706i)14-s + (0.0186 + 0.249i)16-s + (1.18 + 0.364i)17-s + (0.167 − 0.0966i)19-s + (0.189 + 0.830i)20-s + (−0.0244 + 0.106i)22-s + (−0.140 − 0.455i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.732 - 0.681i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.732 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69907 + 1.06160i\)
\(L(\frac12)\) \(\approx\) \(2.69907 + 1.06160i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 \)
7 \( 1 + (0.923 + 2.47i)T \)
good5 \( 1 + (-3.14 - 2.14i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.0766 - 0.508i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-1.07 + 0.856i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-4.86 - 1.50i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-0.729 + 0.421i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.673 + 2.18i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (1.32 - 0.302i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (2.29 + 1.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.31 - 6.78i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (9.45 - 4.55i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (-5.07 - 2.44i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.42 + 11.2i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-7.50 + 8.08i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (5.17 - 3.53i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (3.45 + 3.72i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (1.69 - 2.93i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.06 - 2.06i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (2.42 - 0.953i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (6.41 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.24 - 9.08i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-6.05 - 0.912i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 1.21iT - 97T^{2} \)
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

−10.21401718127602205015151399670, −9.776314793872903883420822995623, −8.454398471786868138743308896630, −7.30929393826734399249746708007, −6.71391260533575119128135492652, −5.93888470303930285998455778824, −5.14216714753186435219899930099, −3.74217226098796843391540786384, −2.96214395353372890272926315533, −1.64763893368945090818972402162, 1.38160716464574692198530089301, 2.38921383412333523428970781356, 3.57925265259186705621556376952, 4.98276957697005211143153300791, 5.63579064653959481019554282042, 6.07363690293918980048751136956, 7.35041265016562082058192738304, 8.707159535414346245238940162653, 9.252846124954999664434286343416, 9.946413414001273699947800265337

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