↑

Properties

Label 2-882-63.41-c1-0-4
Degree $2$
Conductor $882$
Sign $-0.222 + 0.974i$
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 + (−0.0569 + 1.73i)3-s + (0.499 − 0.866i)4-s + (−1.82 + 3.15i)5-s + (−0.816 − 1.52i)6-s + 0.999i·8-s + (−2.99 − 0.197i)9-s − 3.64i·10-s + (−4.38 + 2.53i)11-s + (1.47 + 0.914i)12-s + (2.94 + 1.69i)13-s + (−5.35 − 3.33i)15-s + (−0.5 − 0.866i)16-s + 1.54·17-s + (2.69 − 1.32i)18-s + 0.816i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.0328 + 0.999i)3-s + (0.249 − 0.433i)4-s + (−0.814 + 1.41i)5-s + (−0.333 − 0.623i)6-s + 0.353i·8-s + (−0.997 − 0.0656i)9-s − 1.15i·10-s + (−1.32 + 0.763i)11-s + (0.424 + 0.264i)12-s + (0.816 + 0.471i)13-s + (−1.38 − 0.860i)15-s + (−0.125 − 0.216i)16-s + 0.375·17-s + (0.634 − 0.312i)18-s + 0.187i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.222 + 0.974i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.222 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.253317 - 0.317742i\)
\(L(\frac12)\) \(\approx\) \(0.253317 - 0.317742i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0569 - 1.73i)T \)
7 \( 1 \)
good5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.38 - 2.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 - 0.816iT - 19T^{2} \)
23 \( 1 + (1.47 + 0.850i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.60 - 2.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.87 + 1.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.79T + 37T^{2} \)
41 \( 1 + (-1.01 + 1.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.06 - 5.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.37 - 5.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 + (-1.08 + 1.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 - 3.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.22 - 2.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.74iT - 71T^{2} \)
73 \( 1 - 4.35iT - 73T^{2} \)
79 \( 1 + (6.37 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.768 - 1.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (5.59 - 3.23i)T + (48.5 - 84.0i)T^{2} \)
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

−10.67084498411990376048945986581, −10.01166059456792655746523208151, −9.181092156333121636055975096712, −8.028624305573546083403690712120, −7.57801824675029856148385856044, −6.52982565457459306623092495351, −5.63614012768584184760252923582, −4.43892210084104343976951525752, −3.44514707910203194490080830107, −2.43802049856055631032721486056, 0.26178222458222889117717187497, 1.22589487544506202286535523453, 2.71173953068526014655547752872, 3.88534031299331066816953235090, 5.27824074027313739020188376281, 5.98045410422800835531283797482, 7.49330220410334602425488286572, 7.927608583335547157735674989142, 8.548399625194605586976461510437, 9.213758489926103304691304182719

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